logic and problem solving

Logic Problems

Why logic problems.

A logic problem is a general term for a type of puzzle that is solved through deduction. Given a limited set of truths and a question, we step through the different scenarios until an answer is found. While these problems rarely involving coding, they require problem-solving and the ability to articulate plausible outcomes.

You may encounter logic problems during technical interviews for a programming position so it’s worth developing a strategy on how to approach these questions. They’re also a fun way to strengthen your algorithmic reasoning skills!

Our Question: Apples, Oranges, or Both?

We’ll start with the following problem. You’re faced with three jars labeled “Apples”, “Oranges”, and “Both”. You cannot see the contents of these jars, but you’re informed that each is mislabeled . The contents of the jar are not decribed by the label.

How many times would you need to draw from a jar in order to accurately label each jar?

Our Solution: Using the Information

Let’s boil down our problem into factual statements we can use to draw conclusions.

• The jars are mislabeled
• There are three jars
• One jar is a combination of the contents of the other two jars.

We can use these facts to deduce further information which will be instrumental in solving the problem.

First we can rephrase “the jars are mislabeled” as “the jar labeled ‘Apples’ does not contain Apples “. It’s the same information but presented in a way that will be easier to work towards a solution.

We should also be drawn to item 3: there’s more information available in the “Both” jar which makes it a more “fruitful” source of inquiry. In general, be aware of any exceptions or abnormalities in the phrasing of the question.

Our Solution: Filling in Scenarios

Now we’ll begin walking through hypothetical situations. Being able to reason through the problem and articulate your thought process is essential to performing well in a technical interview.

Let’s imagine we draw a fruit from the jar labeled “Apples”. We know this jar doesn’t only contain apples, but we’re faced with two possibilities. We could draw an apple or we could draw an orange. If we drew an apple, we’d know this was the “Both” jar, but what if we drew an orange? Then this jar remains a mystery, either “Both” and we just happened to draw an orange, or it’s purely “Oranges”. We’re still in the dark!

The thought process is the same for drawing from the “Oranges” jar, so now imagine drawing from the “Both” jar. Again, we may draw either type of fruit, but we’ve learned something more substantial. If we draw an orange, we know this is “Oranges”. If we draw an apple, we know this is “Apples”. There is no ambiguity because the jar is mislabeled as “Both”.

Our Solution: Drawing Conclusions

We’ve identified one jar, do we need to make additional queries? We should return to our Use the Information step. Let’s say we’ve identified “Oranges”.

We have the old “Both” jar, now correctly labeled “Oranges”, and two mislabeled jars: “Oranges” and “Apples”. Can we draw further conclusions? We can!

“Apples” and “Oranges” are both mislabeled, but we have new information. We know where the true “Oranges” is. This doesn’t help us with the mislabeled “Oranges”, it could be either “Both” or “Apples”.

It does help with “Apples”. We know “Apples” is not “Oranges” because we’ve already identified “Oranges”. We also know “Apples” isn’t really “Apples” because it’s mislabeled. That leaves only one option, this jar is “Both”.

With two correctly labeled jars, the third is easily identified as “Apples”

To wrap it up: “Both” –> “Oranges”, which leads us to “Apples” –> “Both” and “Oranges” –> “Apples”

Practice Makes Perfect!

We’ll finish this article with a few practice problems for you to try on your own. Each problem has a link which will take you to an explanation of the solution.

Knights and Knaves

“Knights and knaves” are a popular type of logic puzzle that involves an island inhabited by two types of people: knights and knaves.

• Knights always tell the truth
• Knaves always lie

On the island, you encounter three people, Ted, Ben and Lil.

Ted says, “at least one of the following is true, that Lil is a knave or that I am a knight.”

Ben says, “Ted could claim that I am a knave.”

Lil says, “neither Ted nor Ben are knights.”

Who is a knight and who is a knave?

Here’s the solution.

Three Fastest Horses

We’d like to find the three fastest horses from a group of 25.

We have no stopwatch and our race track has only 5 lanes. No more than 5 horses can be raced at once.

How many races are necessary to evaluate the 3 fastest horses?

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Logical Reasoning | Definition, Strategies & Examples

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Tanya has taught for 21 years, anywhere from 1st through 9th grades, as well as STEM. She has a bachelor's in elementary education with a middle school math endorsement from Oklahoma Wesleyan University. She has a current professional teaching license and years of experience creating interesting, engaging lessons for her students.

Sierra has a Bachelor of Science in mathematics and a Master of Arts in Teaching. She has taught high school math for three years.

What is logical reasoning and why is it important?

Logical reasoning is thinking through a situation and reaching a logical conclusion. It is important because it helps solve problems both big and small.

What is an example of logical reasoning?

Logical reasoning can be used for complex situations. Logical reasoning can also be as simple as concluding that if it is storming outside, then you should not have a picnic lunch.

What is an example of a logic problem?

Transitivity is one type of logical reasoning. An example of a problem using this would be having a statement such as "Tony is older than Sarah and Sarah is older than Jimmy." From that statement, one can reason that Tony is older than Jimmy.

What are logical problems?

Logical problems are ones that can reach a conclusion using a logical, thought-out process. Logical problem-solving strategies can be used to solve these problems.

What are the types of logical reasoning?

There are many different types of logical reasoning. Some basic types of logical reasoning are if/then statements, and/or statements, and transitivity.

Table of Contents

What is logical reasoning, logical problem-solving, how to use logic to solve a problem, examples of using logic to solve a problem, applications of logical problem-solving, lesson summary.

A definition for logical reasoning is thinking through a situation to reach a conclusion or logical consequence. Depending on the situation or clues given, a conclusion is reached that satisfies all aspects of the statement. For example, if the statement is everything outside is wet because it is raining and a person realizes he left his shoes outside, logical reasoning would reach the conclusion that his shoes are wet: His shoes are outside. Everything outside is wet due to the rain. Therefore his shoes are wet.

What are the Types of Logical Reasoning?

There are various types of logical reasoning. Some of the basic types of logical reasoning are if/then statements , and/or statements , and transitivity . Each is briefly described below.

• If/Then Statement: This is also called a '' conditional statement ''. The if part of the statement is the hypothesis while the then part of the statement is the conclusion. If something happens, then this is the result.
• And/Or Statement: And is a conjunction that combines two or more things. Therefore, if a statement includes and both (or all) parts of the statement need to happen or be true in order for the entire statement to be true. With or there is the possibility of different things, but only one will happen. Determining what the actual conclusion will be usually requires more information than what is given in the statement alone.
• Transitivity: Transitivity connects three objects by stating a connection between the first and second object and the second and third object. If the logic holds true, then there should be a connection between the first and third objects as well.

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The definition of logical problem solving is relatively self-explanatory. Logical reasoning, or logic, is used in order to work through a problem and find a solution. If/then statements, and/or statements, and transitivity can all be used to assist in solving problems and reaching conclusions.

• If/Then Statement: For example, ''If the weather is nice, then I will wash the car.'' Using logical reasoning, it can be concluded that the car will get washed with sunny, clear skies. If it is raining or cloudy and windy, the conclusion can be reached that the car will not be washed. Looking at the statement backward can also give information. If the car was washed then it can be surmised that the weather was nice.
• And/Or Statement: As stated previously, an and statement requires both parts to be true in order for the conclusion to be reached. For example, if the statement is, ''You must clean your room and do your homework before you can go to your friend's house'' then both actions must be done. If the child only cleans her room or only does her homework, that is not enough. Both chores have to be done before going to a friend's house can happen. So if the child did not go to her friend's house, the conclusion can be reached that one or both of the chores did not get done. With an or statement a final conclusion cannot always be reached. For example, ''We will have pizza or tacos for dinner'', narrows down the options to two, but is inconclusive about which it will be. It can be concluded that hamburgers will NOT be for dinner, nor will both pizza and tacos. Which one it will be is unknown unless more information is given.
• Transitivity: Transitivity connects two items or concepts that originally were not related. For example, ''Joey is taller than Sandra. Sandra is taller than Hayden.'' Even though the relationship between the size of Joey and Hayden is not stated, logical reasoning can conclude that Joey is taller than Hayden. If Sandra is taller than Hayden, and Joey is taller than Sandra, it must be true that Joey is also taller than Hayden.

With a simple statement such as, ''If today is Saturday, then I will go to the beach'' there are a variety of conclusions that can, and cannot, be drawn from it. The table shows some of those possibilities.

Logic can be used to solve a variety of problems. Take this problem, for example.

Four friends - Gracie, Carolyn, Meghan, and Maria - went shopping for new shoes. The amounts they spent were $42, $46, $50, and $54. What can be logically concluded from the following statement: Gracie spent $4 more than Maria.

First, consider the amount of money that Gracie spent. Of the amounts given, it can be concluded that Gracie could NOT have spent $42 because it is not more than another amount. She spent more than Maria, so $42 is not possible. Next, consider the amount of money that Maria spent. Looking at the options, it can be concluded that Maria did NOT spend $54, because Gracie spent more than Maria and that was the highest amount of money given. The logical conclusions that can be made from that statement, therefore, are:

Gracie did not spend $42.

Maria did not spend $54.

Using logic to solve a problem does not necessarily mean the problem must be difficult and complex. Many times it is a simple situation, but still involves using logical reasoning. For example, say a person finds a young abandoned kitten on the way home. It looks weak and hungry, so what should it be fed? Well, logical reasoning says a kitten is a mammal. Mammals feed their young milk from the mother. Therefore, the kitten could be given milk to drink.

How about the common problem of what to make for dinner? On the drive home from work, a person decides that either pasta or a salad sounds good to eat. More information is needed, though, in order to draw a conclusion. Once the person arrives home, they look through their kitchen to discover they have everything they need to make pasta but do not have any lettuce for a salad. The conclusion can then be reached to make pasta for dinner.

Many fields use logical problem-solving on the job. The medical field is one career that uses logical problem-solving all the time. Doctors examine patients and diagnose problems based on symptoms and logical processes. Engineers use problem-solving as they create and test products. Going through the design process oftentimes brings up issues that need to be fixed. Engineers use logic to determine what the problem is and how to solve it. Construction uses logical reasoning all the time. If the project is built in this specific way, then it will be sturdy and last for years. Mathematical fields use logical reasoning to solve problems, and scientists use problem-solving to work through experiments and draw conclusions.

Logical problem solving is used often in daily life, to the point most probably do not even realize they are doing it. For example:

• Trying to persuade someone can involve logical reasoning. Using logic to draw reasonable conclusions can be much more effective than trying to persuade based solely on emotion.
• When debating between different products to use, logical reasoning is involved. The person knows the outcome or consequence they would like from this product, so items are compared and conclusions reached depending on how well the product will give the desired outcome. For example, which wrinkle cream should I buy? If one has more active ingredients than the other, it could be concluded that it might work better. If shopping online and the product has a lot of positive reviews that show good results, it can be concluded that the product will probably work well.
• When driving and deciding on a route to take, logical reasoning is often a part of that. For example, ''Yesterday there was construction on Main Street and on Broadway. Today I should take the back roads to get to work faster.''

Logical reasoning is thinking through a situation in order to reach a conclusion or logical consequence. Depending on the situation or clues given, a conclusion is reached that satisfies all aspects of the statement. Three basic types of logical reasoning are if/then statements , and/or statements , and transitivity . If/then statements are conditional statements; if something happens, then something else happens. These statements need to be considered forward and backward to determine all possible outcomes. And/or statements put conditions on the situation. ''And'' statements require two or more things to happen before a conclusion is reached, whereas ''or'' statements give two options that could happen but usually require more information before a conclusion can be drawn. Transitivity connects three objects. First, it compares the first to the second, then the second to the third. A conclusion can then be reached about the first and third objects.

Logical reasoning can be used to solve many different types of problems regardless if they are simple or complex. Logical problem solving is used often in daily life. Many careers use logical reasoning, as well, to solve problems in their field.

'' Danger lies before you, while safety lies behind,

Two of us will help you, whichever you would find,... ''

If you are a J.K. Rowling fan you might recognize this line from Harry Potter and the Sorcerer's Stone . Perhaps you've encountered a similar riddle while watching Sherlock or reading an Agatha Christie novel. Logic pervades detective dramas but when looking at logic problems out of context, we can get mixed up between what is truly logical and what is not. In this lesson, I will give you some strategies to help keep it all straight.

The Basics of Logic

The basic building block of logic is an if-then statement such as If the plane is delayed, then I will miss my connection . Depending on the situation or clue given, there are only a couple of logical conclusions that can be drawn. Using straightforward thinking, if the plane was delayed, the logical conclusion is that I missed my connection. However, looking at it backwards can give us an additional conclusion. Say the second part is false and I did NOT miss my connection. The second part of the if-then statement is only false when the first part is, thus the conclusion is that the plane was NOT delayed.

Playing around with parts being true or false can lead to some misconceptions, so let's look at another example and play around. There are four different situations that can happen, but only some of them lead to a logical conclusion. I organized these situations into the table below.

Beyond Basics

Now that we've covered the basics here are some other situations you might encounter in logic problems:

• This makes two pieces of information true at the same time. For example: If I finish my homework AND clean my room, then I will go to the party. If our clue just says I cleaned my room, that is not enough information to conclude I went to the party. Furthermore, if I did NOT go to the party one of those things failed to happen, but we cannot assume which one.
• This narrows down choices, but we won't be able to know for sure which one is true until more information comes along. For example: The cat will play with her catnip OR her mouse toy. We can rule out any other toys that were possible, but won't know more until there's another clue. Furthermore, only one OR the other is possible, NOT BOTH. Because of this, we can also conclude the mouse toy doesn't contain catnip.
• This connects two items that were not originally discussed together. For example: Joey is older than Sam and Sam is older than Luisa. We can conclude Joey has to be older than Luisa too, even though we weren't explicitly told so.
• As described above we sometimes look at parts of clues NOT being true. If you encounter a double negative (a not not) simplify it and take the double negative out. For example: If it is sunny outside, then I will not bring an umbrella. We wouldn't say I am not not bringing an umbrella; it would simply be, I am bringing an umbrella. Be careful to only eliminate true double negatives. Don't say that being not sunny means it's raining. It could actually be snowing or hailing and your conclusion is incorrect.

Process of Elimination

When solving logic problems it's more than likely you'll reach a conclusion just because no other possibilities are left. For our last example we'll consider the type of problem where you are matching different collections together. This could be people to planes to arrival times, or prices to fruit, etc. In these problems, all options are given at the start so you can mark out what's not possible.

Example: Five people arrived five minutes apart for a movie. The times were 11:00, 11:05, 11:10, 11:15, and 11:20.

Clue: Mandy arrived five minutes after Jake.

Conclusion: Jake did NOT arrive at 11:20 and Mandy did NOT arrive at 11:00

• Turn this into an if-then statement: If Jake arrives at _____, then Mandy arrives at _____ +5 minutes.
• Test out the times.
• If Jake arrives at 11:20, we would conclude Mandy arrived at 11:25, which is not an option.
• Since the second part is false, the first part must be as well, so Jake did NOT arrive at 11:20.
• Flipping it around, If Mandy arrives at _____, then Jake arrives at _____-5 minutes.
• If Mandy arrived at 11:00 that means Jake arrived at 10:55, also not an option.

When you're solving logic problems the most important thing is to never assume too much. When given if-then statements think backwards too. Look at what you can conclude when the then part is false. Don't forget an and requires both parts to be true but ors mean one or the other and not both. When two items are compared to the same thing, look for transitivity to connect them to each other. Finally, process of elimination might seem like the long way, but it's a powerful tool.

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How to learn logical reasoning for coding and beyond

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From the binary digits that computers use to represent data, to the most sophisticated AI programs, logic is key to the development and implementation of the technologies we rely on. Logic isn’t just a tool for programmers, though. We all use logical reasoning on a daily basis to think through problems and make decisions.

There are a lot of advantages to burnishing your logic skills. If you’re learning to code , practicing logical reasoning will help you understand how programming languages work and how programmers use them to solve problems. If you’re not at that stage yet, that’s all right: learning logical reasoning will make you a more effective and self-aware thinker , and that has applications far beyond programming.

Today, we’re going to discuss logical reasoning: what it is, why it’s important in the context of programming, and how you can start learning it.

We’ll cover :

Logical reasoning defined

2 types of logical reasoning, how programmers use logical reasoning, strategies for practicing logical reasoning, next steps for learning logical reasoning.

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Narrowly defined, logical reasoning is the practice of applying induction, deduction, or another logical method to a problem. More broadly, using logical reasoning means analyzing the relationships among the constituent parts of an argument or process . It involves thinking about how elements interact to bring about a certain result.

Logical reasoning has tons of practical applications. You can use it to construct strong arguments and analyze the arguments of others. You can use it to develop step-based processes that are efficient, effective, and internally coherent. This is why logical reasoning is so important to so many disciplines. Philosophers, scientists, mathematicians, and computer programmers all use logic in their work.

There are many types of logical reasoning. We’ll discuss two of the most important before exploring the role of logical reasoning in programming, specifically.

Inductive reasoning involves evaluating a body of information to derive a general conclusion . Whenever you engage in research, you’re using inductive reasoning insofar as you’re gathering evidence and using it to better understand the topic at hand. It’s important to remember, however, that the conclusions you’ll arrive at through induction will often be built on evidence that is incomplete. This means that the outcome of inductive reasoning tends to be probable rather than certain.

Deductive reasoning is the process of deriving conclusions from general statements or principles called premises . The syllogism is one of the most basic forms of deductive reasoning. A syllogism includes a major premise, a minor premise, and a conclusion. Here’s an example:

“All bears have fur. A polar bear is a type of bear. Therefore, polar bears have fur.”

In this case, the premises are true, and they lead logically to the conclusion. A deductive argument can easily go wrong if the conclusion doesn’t follow logically from the premises or if one or more of the premises are false. The following syllogism is flawed:

“All bears live in forests. A polar bear is a type of bear. Therefore, polar bears live in forests.”

The major premise is incorrect, so the conclusion is also incorrect.

In summary, while induction involves deriving general principles from specific cases, deduction involves applying general principles to specific cases.

Logical reasoning plays an important role in programming. Let’s discuss some of the ways that programmers use logic in their work.

Problem-solving

Programmers use logical reasoning for problem-solving . Before you start coding a program, you’ll have to wrestle with questions about what the program is trying to accomplish, what features it will need to have, what programming language you’ll write it in, and so on. Inductive and deductive reasoning are useful tools for answering process questions like these.

Writing code

Coding is the process of writing instructions in a language a computer can read. These instructions direct the computer to perform a set of operations . Writing functioning code requires applying logic. Imperative programming , for instance, is a paradigm based on issuing commands for a program to follow on a step-by-step basis. Writing these commands is a logical process. It requires thinking about the relationships between parts. Do the commands interact with each other to bring about the desired outcome? If not, the code isn’t going to work.

Writing functioning code is also dependent on understanding the logic of the programming language you’re using. For example, the logical operators and, or, and not in Python are derived from Boolean logic , and they’re an important part of conditional statements and other logical expressions.

Creating algorithms

Algorithms used in machine learning and artificial intelligence also rely on logic. An algorithm is a set of instructions that tells a program what to do on the basis of certain inputs. A simple decision tree algorithm, for instance, will consist of a series of branching decisions. The logic of each branch is fairly simple: “If a certain condition is met, do A . If not, do B .” These branching decisions are repeated through later stages of the program to specify the appropriate end result based on the inputs.

The best way to practice logical reasoning is by consciously implementing it in your daily life. Here are a few exercises that might be helpful.

Find a news article or thinkpiece in a reputable publication. Read it and try to pick out the author’s primary claim . Then, try to identify the pieces of evidence that the author uses to support that claim. Does the author’s claim follow logically from the evidence? Or does the claim go beyond what the evidence actually supports?

Analyzing how authors use evidence to support their arguments is a great way of practicing inductive reasoning.

Next time you’re planning how to tackle a new project, break your process down into steps and then write them down on paper. Does the end result follow logically from the steps that precede it? Or does the process contain steps that are redundant, unnecessary, or counterproductive?

Remember that logic is all about coherence among the constituent elements of an argument or process. Thinking logically means thinking holistically about how relationships and interactions lead to a certain result.

Find an issue being presented for discussion at the next meeting of your local City Council or zoning board. Formulate a position on it, and use logical reasoning to construct an argument that articulates and defends that position .

As we discussed earlier, logical reasoning is a great tool for analyzing others’ arguments, but you can also use it to construct sound arguments of your own. The best way to practice this is by taking a position on an issue and trying to convince others that you’re right.

There’s a good chance you already use the forms of logical reasoning we’ve described in this article. We arrive at judgments and decisions by evaluating evidence (induction) and applying principles that we believe to be true (deduction). Even so, studying logical reasoning and consciously applying it can make you a more rigorous and self-aware thinker. It will help you to pick out flaws in your logic and others’ , and it will help you to solve problems and build more efficient processes – in programming work and beyond.

If you’re interested in learning more about how programmers think, consider exploring Educative’s learn to code courses.

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They provide an introduction to programming and how programmers solve problems. It also offers some insight into the logical structure of programming languages, explaining, for instance, how conditional statements work to ensure that programs perform the right operations in response to inputs.

Whether you’re learning to code or just trying to strengthen your problem-solving skills, you’ll find that the rigorous, step-by-step thinking that logical reasoning requires is a major asset.

Happy learning!

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Absolute beginner’s guide to computers and programming

How to think like a programmer: 3 misconceptions debunked

How much math do you need to know to be a developer?

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How can I practice my logic?

To practice logic in coding, try the following:

• Solve challenges on platforms like LeetCode or Educative.
• Build small projects.
• Debug and fix code errors.
• Study data structures.
• Participate in coding contests.
• Collaborate in peer programming.
• Learn different programming paradigms.
• Read and analyze open-source code.

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How to think like a programmer — lessons in problem solving

by Richard Reis

If you’re interested in programming, you may well have seen this quote before:

“Everyone in this country should learn to program a computer, because it teaches you to think.” — Steve Jobs

You probably also wondered what does it mean, exactly, to think like a programmer? And how do you do it??

Essentially, it’s all about a more effective way for problem solving .

In this post, my goal is to teach you that way.

By the end of it, you’ll know exactly what steps to take to be a better problem-solver.

Why is this important?

Problem solving is the meta-skill.

We all have problems. Big and small. How we deal with them is sometimes, well…pretty random.

Unless you have a system, this is probably how you “solve” problems (which is what I did when I started coding):

• Try a solution.
• If that doesn’t work, try another one.
• If that doesn’t work, repeat step 2 until you luck out.

Look, sometimes you luck out. But that is the worst way to solve problems! And it’s a huge, huge waste of time.

The best way involves a) having a framework and b) practicing it.

“Almost all employers prioritize problem-solving skills first.

Problem-solving skills are almost unanimously the most important qualification that employers look for….more than programming languages proficiency, debugging, and system design.

Demonstrating computational thinking or the ability to break down large, complex problems is just as valuable (if not more so) than the baseline technical skills required for a job.” — Hacker Rank ( 2018 Developer Skills Report )

Have a framework

To find the right framework, I followed the advice in Tim Ferriss’ book on learning, “ The 4-Hour Chef ”.

It led me to interview two really impressive people: C. Jordan Ball (ranked 1st or 2nd out of 65,000+ users on Coderbyte ), and V. Anton Spraul (author of the book “ Think Like a Programmer: An Introduction to Creative Problem Solving ”).

I asked them the same questions, and guess what? Their answers were pretty similar!

Soon, you too will know them.

Sidenote: this doesn’t mean they did everything the same way. Everyone is different. You’ll be different. But if you start with principles we all agree are good, you’ll get a lot further a lot quicker.

“The biggest mistake I see new programmers make is focusing on learning syntax instead of learning how to solve problems.” — V. Anton Spraul

So, what should you do when you encounter a new problem?

Here are the steps:

1. Understand

Know exactly what is being asked. Most hard problems are hard because you don’t understand them (hence why this is the first step).

How to know when you understand a problem? When you can explain it in plain English.

Do you remember being stuck on a problem, you start explaining it, and you instantly see holes in the logic you didn’t see before?

Most programmers know this feeling.

This is why you should write down your problem, doodle a diagram, or tell someone else about it (or thing… some people use a rubber duck ).

“If you can’t explain something in simple terms, you don’t understand it.” — Richard Feynman

Don’t dive right into solving without a plan (and somehow hope you can muddle your way through). Plan your solution!

Nothing can help you if you can’t write down the exact steps.

In programming, this means don’t start hacking straight away. Give your brain time to analyze the problem and process the information.

To get a good plan, answer this question:

“Given input X, what are the steps necessary to return output Y?”

Sidenote: Programmers have a great tool to help them with this… Comments!

Pay attention. This is the most important step of all.

Do not try to solve one big problem. You will cry.

Instead, break it into sub-problems. These sub-problems are much easier to solve.

Then, solve each sub-problem one by one. Begin with the simplest. Simplest means you know the answer (or are closer to that answer).

After that, simplest means this sub-problem being solved doesn’t depend on others being solved.

Once you solved every sub-problem, connect the dots.

Connecting all your “sub-solutions” will give you the solution to the original problem. Congratulations!

This technique is a cornerstone of problem-solving. Remember it (read this step again, if you must).

“If I could teach every beginning programmer one problem-solving skill, it would be the ‘reduce the problem technique.’

For example, suppose you’re a new programmer and you’re asked to write a program that reads ten numbers and figures out which number is the third highest. For a brand-new programmer, that can be a tough assignment, even though it only requires basic programming syntax.

If you’re stuck, you should reduce the problem to something simpler. Instead of the third-highest number, what about finding the highest overall? Still too tough? What about finding the largest of just three numbers? Or the larger of two?

Reduce the problem to the point where you know how to solve it and write the solution. Then expand the problem slightly and rewrite the solution to match, and keep going until you are back where you started.” — V. Anton Spraul

By now, you’re probably sitting there thinking “Hey Richard... That’s cool and all, but what if I’m stuck and can’t even solve a sub-problem??”

First off, take a deep breath. Second, that’s fair.

Don’t worry though, friend. This happens to everyone!

The difference is the best programmers/problem-solvers are more curious about bugs/errors than irritated.

In fact, here are three things to try when facing a whammy:

• Debug: Go step by step through your solution trying to find where you went wrong. Programmers call this debugging (in fact, this is all a debugger does).

“The art of debugging is figuring out what you really told your program to do rather than what you thought you told it to do.”” — Andrew Singer

• Reassess: Take a step back. Look at the problem from another perspective. Is there anything that can be abstracted to a more general approach?

“Sometimes we get so lost in the details of a problem that we overlook general principles that would solve the problem at a more general level. […]

The classic example of this, of course, is the summation of a long list of consecutive integers, 1 + 2 + 3 + … + n, which a very young Gauss quickly recognized was simply n(n+1)/2, thus avoiding the effort of having to do the addition.” — C. Jordan Ball

Sidenote: Another way of reassessing is starting anew. Delete everything and begin again with fresh eyes. I’m serious. You’ll be dumbfounded at how effective this is.

• Research: Ahh, good ol’ Google. You read that right. No matter what problem you have, someone has probably solved it. Find that person/ solution. In fact, do this even if you solved the problem! (You can learn a lot from other people’s solutions).

Caveat: Don’t look for a solution to the big problem. Only look for solutions to sub-problems. Why? Because unless you struggle (even a little bit), you won’t learn anything. If you don’t learn anything, you wasted your time.

Don’t expect to be great after just one week. If you want to be a good problem-solver, solve a lot of problems!

Practice. Practice. Practice. It’ll only be a matter of time before you recognize that “this problem could easily be solved with <insert concept here>.”

How to practice? There are options out the wazoo!

Chess puzzles, math problems, Sudoku, Go, Monopoly, video-games, cryptokitties, bla… bla… bla….

In fact, a common pattern amongst successful people is their habit of practicing “micro problem-solving.” For example, Peter Thiel plays chess, and Elon Musk plays video-games.

“Byron Reeves said ‘If you want to see what business leadership may look like in three to five years, look at what’s happening in online games.’

Fast-forward to today. Elon [Musk], Reid [Hoffman], Mark Zuckerberg and many others say that games have been foundational to their success in building their companies.” — Mary Meeker ( 2017 internet trends report )

Does this mean you should just play video-games? Not at all.

But what are video-games all about? That’s right, problem-solving!

So, what you should do is find an outlet to practice. Something that allows you to solve many micro-problems (ideally, something you enjoy).

For example, I enjoy coding challenges. Every day, I try to solve at least one challenge (usually on Coderbyte ).

Like I said, all problems share similar patterns.

That’s all folks!

Now, you know better what it means to “think like a programmer.”

You also know that problem-solving is an incredible skill to cultivate (the meta-skill).

As if that wasn’t enough, notice how you also know what to do to practice your problem-solving skills!

Phew… Pretty cool right?

Finally, I wish you encounter many problems.

You read that right. At least now you know how to solve them! (also, you’ll learn that with every solution, you improve).

“Just when you think you’ve successfully navigated one obstacle, another emerges. But that’s what keeps life interesting.[…]

Life is a process of breaking through these impediments — a series of fortified lines that we must break through.

Each time, you’ll learn something.

Each time, you’ll develop strength, wisdom, and perspective.

Each time, a little more of the competition falls away. Until all that is left is you: the best version of you.” — Ryan Holiday ( The Obstacle is the Way )

Now, go solve some problems!

And best of luck ?

Special thanks to C. Jordan Ball and V. Anton Spraul . All the good advice here came from them.

Thanks for reading! If you enjoyed it, test how many times can you hit in 5 seconds. It’s great cardio for your fingers AND will help other people see the story.

If this article was helpful, share it .

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Logical Puzzles

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• Andrew Hayes

A logical puzzle is a problem that can be solved through deductive reasoning. This page gives a summary of the types of logical puzzles one might come across and the problem-solving techniques used to solve them.

Elimination Grids

Truth tellers and liars, cryptograms, arithmetic puzzles, river crossing puzzle, tour puzzles, battleship puzzles, chess puzzles, k-level thinking, other puzzles.

Main Article: Propositional Logic See Also: Predicate Logic

One of the simplest types of logical puzzles is a syllogism . In this type of puzzle, you are given a set of statements, and you are required to determine some truth from those statements. These types of puzzles can often be solved by applying principles from propositional logic and predicate logic . The following syllogism is from Charles Lutwidge Dodgson, better known under his pen name, Lewis Carroll.

I have a dish of potatoes. The following statements are true: No potatoes of mine, that are new, have been boiled. All my potatoes in this dish are fit to eat. No unboiled potatoes of mine are fit to eat. Are there any new potatoes in this dish? The first and third statements can be connected by a transitive argument. All of the new potatoes are unboiled, and unboiled potatoes aren't fit to eat, so no new potatoes are fit to eat. The second statement can be expressed as the equivalent contrapositive. All of the potatoes in the dish are fit to eat; if there is a potato that is not fit to eat, it isn't in the dish. Then, once again, a transitive argument is applied. New potatoes aren't fit to eat, and inedible potatoes aren't in the dish. Thus, there are no new potatoes in the dish. \(_\square\)

Given below are three statements followed by three conclusions. Take the three statements to be true even if they vary from commonly known facts. Read the statements and decide which conclusions follow logically from the statements.

Statements: 1. All actors are musicians. 2. No musician is a singer. 3. Some singers are dancers.

Conclusions: 1. Some actors are singers. 2. Some dancers are actors. 3. No actor is a singer.

Answer Choices: a) Only conclusion 1 follows. b) Only conclusion 2 follows. c) Only conclusion 3 follows. d) At least 2 of the conclusions follows.

Main Article: Elimination Grids

Some logical puzzles require you to determine the correct pairings for sets of objects. These puzzles can often be solved with the process of elimination, and an elimination grid is an effective tool to apply this process.

An example of an elimination grid

Elimination grids are aligned such that each row represents an object within a set, and each column represents an object to be paired with an object from that set. Check marks and X marks are used to show which objects pair, and which objects do not pair.

Mr. and Mrs. Tan have four children--three boys and a girl-- who each like one of the colors--blue, green, red, yellow-- and one of the letters--P, Q, R, S.

• The oldest child likes the letter Q.
• The youngest child likes green.
• Alfred likes the letter S.
• Brenda has an older brother who likes R.
• The one who likes blue isn't the oldest.
• The one who likes red likes the letter P.
• Charles likes yellow.

Based on the above facts, Darius is the \(\text{__________}.\)

Main Article: Truth-Tellers and Liars

A variation on elimination puzzles is a truth-teller and liar puzzle , also known as a knights and knaves puzzle . In this type of puzzle, you are given a set of people and their respective statements, and you are also told that some of the people always tell the truth and some always lie. The goal of the puzzle is to deduce the truth from the given statements.

20\(^\text{th}\) century mathematician Raymond Smullyan popularized these types of puzzles.

You are in a room with three chests. You know at least one has treasure, and if a chest has no treasure, it contains deadly poison.

Each chest has a message on it, but all the messages are lying .

• Left chest: "The middle chest has treasure."
• Middle chest: "All these chests have treasure."
• Right chest: "Only one of these chests has treasure."

Which chests have treasure?

There are two people, A and B , each of whom is either a knight or a knave.

A says, "At least one of us is a knave."

What are A and B ?

\(\) Details and Assumptions:

• A knight always tells the truth.
• A knave always lies.

Main Article: Cryptograms

A cryptogram is a puzzle in which numerical digits in a number sentence are replaced with characters, and the goal of the puzzle is to determine the values of these characters.

\[ \begin{array} { l l l l l } & &P & P & Q \\ & &P & Q & Q \\ + && Q & Q & Q \\ \hline & & 8 & 7 & 6 \\ \end{array} \]

In the sum shown above, \(P\) and \(Q\) each represent a digit. What is the value of \(P+Q\)?

\[ \overline{EVE} \div \overline{DID} = 0. \overline{TALKTALKTALKTALK\ldots} \]

Given that \(E,V,D,I,T,A,L\) and \(K \) are distinct single digits, let \(\overline{EVE} \) and \( \overline{DID} \) be two coprime 3-digit positive integers and \(\overline{TALK} \) be a 4-digit integer, such that the equation above holds true, where the right hand side is a repeating decimal number.

Find the value of the sum \( \overline{EVE} + \overline{DID} + \overline{TALK} \).

Main Articles: Fill in the Blanks and Operator Search

Arithmetic puzzles contain a series of numbers, operations, and blanks in order, and the object of the puzzle is to fill in the blanks to obtain a desired result.

\[\huge{\Box \times \Box \Box = \Box \Box \Box}\]

Fill the boxes above with the digits \(1,2,3,4,5,6\), with no digit repeated, such that the equation is true.

Enter your answer by concatenating all digits in the order they appear. For example, if the answer is \(1 \times 23 = 456\), enter \(123456\) as your final answer.

Also try its sister problem.

\[ \LARGE{\begin{eqnarray} \boxed{\phantom0} \; + \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; - \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; \times \; \boxed{\phantom0}\; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; \div \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \end{eqnarray}} \]

Put one of the integers \(1, 2, \ldots , 13\) into each of the boxes, such that twelve of these numbers are used once for each (and one number is not used at all) and all four equations are true.

What is the sum of all possible values of the missing (not used) number?

Main Article: River Crossing Puzzles

In a river crossing puzzle , the goal is to find a way to move a group of people or objects across a river (or some other kind of obstacle), and to do it in the fewest amount of steps or least amount of time.

A famous river crossing problem is Richard Hovasse's bridge and torch problem , written below.

Four people come to a river in the night. There is a narrow bridge, but it can only hold two people at a time. They have one torch and, because it's night, the torch has to be used when crossing the bridge. Person A can cross the bridge in one minute, B in two minutes, C in five minutes, and D in eight minutes. When two people cross the bridge together, they must move at the slower person's pace. The question is, can they all get across the bridge in 15 minutes or less? Assume that a solution minimizes the total number of crosses. This gives a total of five crosses--three pair crosses and two solo crosses. Also, assume we always choose the fastest for the solo cross. First, we show that if the two slowest persons (C and D) cross separately, they accumulate a total crossing time of 15. This is done by taking persons A, C, D: D+A+C+A = 8+1+5+1=15. (Here we use A because we know that using A to cross both C and D separately is the most efficient.) But, the time has elapsed and persons A and B are still on the starting side of the bridge and must cross. So it is not possible for the two slowest (C and D) to cross separately. Second, we show that in order for C and D to cross together that they need to cross on the second pair cross: i.e. not C or D, so A and B, must cross together first. Remember our assumption at the beginning states that we should minimize crosses, so we have five crosses--3 pair crossings and 2 single crossings. Assume that C and D cross first. But then C or D must cross back to bring the torch to the other side, so whoever solo-crossed must cross again. Hence, they will cross separately. Also, it is impossible for them to cross together last, since this implies that one of them must have crossed previously, otherwise there would be three persons total on the start side. So, since there are only three choices for the pair crossings and C and D cannot cross first or last, they must cross together on the second, or middle, pair crossing. Putting all this together, A and B must cross first, since we know C and D cannot and we are minimizing crossings. Then, A must cross next, since we assume we should choose the fastest to make the solo cross. Then we are at the second, or middle, pair crossing, so C and D must go. Then we choose to send the fastest back, which is B. A and B are now on the start side and must cross for the last pair crossing. This gives us, B+A+D+B+B = 2+1+8+2+2 = 15. It is possible for all four people to cross in 15 minutes. \(_\square\)

Main Article: Tour Puzzles See Also: Eulerian Path

In a tour puzzle , the goal is to determine the correct path for an object to traverse a graph. These kinds of puzzles can take several forms: chess tours, maze traversals, eulerian paths , and others.

Find the path that leads from the star in the center back to the star in the center. Paths can only go in the direction of an arrow. Image Credit: Eric Fisk Show Solution The solution path is outlined in red below.

Determine a path through the below graph such that each edge is traversed exactly once . Show Solution There are several possible solutions. One possible solution is shown below, with the edges marked in the order they are traversed.

A chess tour is an interesting type of puzzle in its own right, and is explained in detail further down the page.

Main Article: Nonograms

A nonogram is a grid-based puzzle in which a series of numerical clues are given beside a rectangular grid. When the puzzle is completed, a picture is formed in the grid.

The puzzle begins with a series of numbers on the left and above the grid. Each of these numbers represents a consecutive run of shaded spaces in the corresponding row or column. Each consecutive run is separated from other runs by at least one empty space. The puzzle is complete when all of the numbers have been satisfied. The primary technique to solve these puzzles is the process of elimination. If the puzzle is designed correctly, there should be no guesswork required.

Complete the nonogram: Show Solution

One of the many logical puzzles is the Battleship puzzle (sometimes called Bimaru, Yubotu, Solitaire Battleships or Battleship Solitaire). The puzzle is based on the Battleship game.

Solitaire Battleships was invented by Jaime Poniachik in Argentina and was first featured in the magazine Humor & Juegos.

This is an example of a solved Battleship puzzle. The puzzle consists of a 10 × 10 small squares, which contain the following:

• 1 battleship 4 squares long
• 2 cruisers 3 squares long each
• 3 destroyers 2 squares long each
• 4 submarines 1 square long each.

They can be put horizontally or vertically, but never diagonally. The boats are placed so that no boats touch each other, not even vertically. The numbers beside the row/column indicate the numbers of squares occupied in the row/column, respectively. ⬤ indicates a submarine and ⬛ indicates the body of a ship, while the half circles indicate the beginning/end of a ship.

The goal of the game is to fill in the grid with water or ships.

Main Article: Sudoku

A sudoku is a puzzle on a \(9\times 9\) grid in which each row, column, and smaller square portion contains each of the digits 1 through 9, each no more than once. Each puzzle begins with some of the spaces on the grid filled in. The goal is to fill in the remaining spaces on the puzzle. The puzzle is solved primarily through the process of elimination. No guesswork should be required to solve, and there should be only one solution for any given puzzle.

Solve the sudoku puzzle: Puzzle generated by Open Sky Sudoku Generator Each row should contain the each of the digits 1 through 9 exactly once. The same is true for columns and the smaller \(3\times 3\) squares. Show Solution

Main Article: Chess Puzzles See Also: Reduced Games , Opening Strategies , and Rook Strategies

Chess puzzles take the rules of chess and challenge you to perform certain actions or deduce board states.

One kind of chess puzzle is a chess tour , related to the tour puzzles mentioned in the section above. This kind of puzzle challenges you to develop a tour of a chess piece around the board, applying the rules of how that piece moves.

Dan and Sam play a game on a \(5\times3\) board. Dan places a White Knight on a corner and Sam places a Black Knight on the nearest corner. Each one moves his Knight in his turn to squares that have not been already visited by any of the Knights at any moment of the match.

For example, Dan moves, then Sam, and Dan wants to go to Black Knight's initial square, but he can't, because this square has been occupied earlier.

When someone can't move, he loses. If Dan begins, who will win, assuming both players play optimally?

This is the seventeenth problem of the set Winning Strategies.

Due to its well-defined ruleset, the game of chess affords many different types of puzzles. The problem below shows that you can even deduce whose turn it is from a certain boardstate (or perhaps you cannot).

Whose move is it now?

Main Article: K-Level Thinking See Also: Induction - Introduction

K-level thinking is the name of a kind of assumption in certain logic puzzles. In these types of puzzles, there are a number of actors in a situation, and each of them is perfectly logical in their decision-making. Furthermore, each of these actors is aware that all other actors in the situation are perfectly logical in their decision-making.

Calvin, Zandra, and Eli are students in Mr. Silverman's math class. Mr. Silverman hands each of them a sealed envelope with a number written inside.

He tells them that they each have a positive integer and the sum of the three numbers is 14. They each open their envelope and inspect their own number without seeing the other numbers.

Calvin says,"I know that Zandra and Eli each have a different number." Zandra replies, "I already knew that all three of our numbers were different." After a brief pause Eli finally says, "Ah, now I know what number everyone has!"

What number did each student get?

Format your answer by writing Calvin's number first, then Zandra's number, and finally Eli's number. For example, if Calvin has 8, Zandra has 12, and Eli has 8, the answer would be 8128.

Two logicians must find two distinct integers \(A\) and \(B\) such that they are both between 2 and 100 inclusive, and \(A\) divides \(B\). The first logician knows the sum \( A + B \) and the second logician knows the difference \(B-A\).

Then the following discussion takes place:

Logician 1: I don't know them. Logician 2: I already knew that.

Logician 1: I already know that you are supposed to know that. Logician 2: I think that... I know... that you were about to say that!

Logician 1: I still can't figure out what the two numbers are. Logician 2: Oops! My bad... my previous conclusion was unwarranted. I didn't know that yet!

What are the two numbers?

Enter your answer as a decimal number \(A.B\). \((\)For example, if \(A=23\) and \(B=92\), write \(23.92.)\)

Note: In this problem, the participants are not in a contest on who finds numbers first. If one of them has sufficient information to determine the numbers, he may keep this quiet. Therefore nothing may be inferred from silence. The only information to be used are the explicit declarations in the dialogue.

Of course, the puzzles outlined above aren't the only types of puzzles one might encounter. Below are a few more logical puzzles that are unrelated to the types outlined above.

You are asked to guess an integer between \(1\) and \(N\) inclusive.

Each time you make a guess, you are told either

(a) you are too high, (b) you are too low, or (c) you got it!

You are allowed to guess too high twice and too low twice, but if you have a \(3^\text{rd}\) guess that is too high or a \(3^\text{rd}\) guess that is too low, you are out.

What is the maximum \(N\) for which you are guaranteed to accomplish this?

\(\) Clarification : For example, if you were allowed to guess too high once and too low once, you could guarantee to guess the right answer if \(N=5\), but not for \(N>5\). So, in this case, the answer would be \(5\).

You play a game with a pile of \(N\) gold coins.

You and a friend take turns removing 1, 3, or 6 coins from the pile. The winner is the one who takes the last coin.

For the person that goes first, how many winning strategies are there for \(N < 1000?\)

\(\) Clarification: For \(1 \leq N \leq 999\), for how many values of \(N\) can the first player develop a winning strategy?

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“In some ways, programming is like a painting. You start with a blank canvas and certain basic raw materials. You use a combination of science, art, and craft to determine what to do with them.” – Andrew Hunt

Yes, programming in itself is a very beautiful art. Sometimes we may face some problems while trying to program, but we can definitely overcome them. So, in this article, we will be sharing the top 15 tips and techniques that can help you to make your programming skills more strong, and rectify some common programming problems and this will also help you in the logic-building process.

How to Improve Your Logic-Building Skills in Programming?

Here are the ways in which you can improve your logic-building skills in programming. So let’s get started!!!

1. Concepts are Building Blocks for Programming

While trying to crack the logic of any coding problem, many of us think that we never came across such algorithms or theorems while studying and therefore are not able to solve the problem. In order to solve any problem, we should know the concepts of that topic, then only we would be able to apply them and solve the problem. Theoretical knowledge and concepts can be gained by reading articles, blogs, documentation, and watching videos based on that topic. You can also refer to the articles on GeeksforGeeks for building your concepts. We should also know the application of concepts and practice some important problems based on that topic.

2. Be Consistent

Many times it happens that we take up a challenge to solve a question for some number of days and then discontinue in the middle after some days!! It is a popular saying that practice makes a man perfect!! The same is the case with building programming logic. Make it a point to revise, or read an article or solve a question daily despite being very busy with remaining activities. Practicing consistently will help a lot in the overall logic-building process. In order to motivate yourself, you should always contemplate the reason why you started, reward yourself, and make programming fun by solving some quizzes and experimenting with the programs to see different outputs.

3. Pen and Paper Approach

After seeing any problem, we generally start coding the same on our IDE. So, when we are asked to write code on paper in interviews, we fail to do so. Always try to write the pseudo code or algorithm of the code before implementing them. It will help you in writing the code and the next time whenever you approach a similar problem you will be able to recollect more easily. It will also help you in getting syntactically strong.

4. Revision is Very Important

Many of you might be facing this issue that you learn a particular concept but after a few days or months when another question with the same logic or concept appears, you are unable to solve it. This is because you haven’t revised the concepts. Always make it a point to write down the important concepts and logic of questions that are important and keep them revised again and again. This will help you in recollecting the concepts easily.

5. Do as Many Questions as You Can 

It happens with most of us that there comes a single question and most of us get stuck there for 4 to 5 days and still are not able to crack it. Always try to practice lots of questions in order to develop your programming logic skills. This will help you in improving your logic building. If you are stuck on a single question, don’t spend a lot of time after a single question instead look for the concepts hidden behind the question.

6. Puzzle Solving

In many coding competitions, problems are not directly asked based on a concept. Instead, it generally involves a story woven around it, and we have to figure out the logic for solving the program. In such cases, sometimes we are unable to solve the problem. Try solving puzzles such as Sudoku to develop your logic and thinking ability because programming is nothing but solving complex problems with the help of good logic. 

7. Follow Step-by-Step Approach

We don’t start running since the day we are born. Similar logic applies to coding also. We should not directly jump to difficult questions. We should go from Basic to Advance questions. You can take the ratio of questions such while choosing 10 questions you can divide them into 5 easy, 3 medium, and 2 hard questions. You can find these questions on many good websites. Sometimes, people solve a lot of easy questions from all the sites, but they are not able to solve medium-level questions. Instead, make a balance of all the levels. This will help in clearing the coding tests while placements as most of the questions are from easy to medium level.

8. Find a Programmer’s Community

Sometimes we get bored while solving problems by ourselves with no one to teach or guide us. In such cases, you can always try discussing solutions or complex questions with fellow programmers and friends. This will always help you in finding new logic for the same problem and will help you in optimizing your code. This will also improve your confidence and communication skills!!

9. Go through the Editorials

It happens a lot of times that we are not able to solve some questions, so we just leave the question or understand the editorial and move forward without implementing it. After programming any question, go through the editorial section and the top submissions of the code. Here you will be able to find optimized and different logic for the same code. Try to implement the solutions in the editorial section after understanding them, so that the next time you find such a question you will be able to solve it.

10. Take Part in Coding Challenges

Most people are aware of coding challenges and if you want to build your logical skills then you must keep taking part in the same. Taking part regularly in coding challenges is very useful as it makes you familiar with the logical mindset. In a coding challenge, there are numerous types of questions that provide you with a lot of exposure. Also, taking part in such challenges allows you to see solutions of various codes provided by different coders and helps you if stuck at some point. 

11. Learn New Things Regularly

Programmers should never stop learning or being stuck on one topic. They must keep on solving multiple topics as it will help them to expand their area of knowledge by building technical skills. The aim should be solving new problems daily and not being stuck to the old pattern or algorithm in order to achieve success. However, at times some topics are a bit tough and take numerous attempts to solve, in that case, stop solving that and go on to the next one as sometimes new problems are helpful in solving the old ones. 

12. Understand Mathematical Concepts

Mathematics is an important aspect of programming and understanding properly will help you in making numerous visuals or graphs, coding in applications, simulation, problem-solving applications, design of algorithms, etc.

13. Build Projects

Project building is another task that will enhance your logical building skills in programming. It challenges your ability to tackling with new things by using different methods and tactics. It is recommended that you must build one project in order to get a proper clarity of the subject and assess yourself in order to work ahead efficiently.

14. Notes Preparation

Notes are saviors and if one does that regularly then nothing can beat them from achieving their goal. While making notes you must write down every trick, concept, and algorithm so that if you need it again it is easily available. So if you are solving any problem then make sure to note down the library functions it will also be helpful for your future interviews. Noting down basic algorithms such as merge sort, binary search, etc. will help you if you are stuck somewhere. 

15. Patience is the Key

Most of the time we leave programming after some days just because we are unable to solve the questions. Let’s always motivate ourselves by saying let’s just try one more time differently, before we decide to quit!!!

If you’ll patiently work on your programming logic skills and follow the tips which we have shared with you, no one can stop you from being a good programmer and you will surely crack all the coding tests and interviews!!!

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How to master the seven-step problem-solving process

In this episode of the McKinsey Podcast , Simon London speaks with Charles Conn, CEO of venture-capital firm Oxford Sciences Innovation, and McKinsey senior partner Hugo Sarrazin about the complexities of different problem-solving strategies.

Podcast transcript

Simon London: Hello, and welcome to this episode of the McKinsey Podcast , with me, Simon London. What’s the number-one skill you need to succeed professionally? Salesmanship, perhaps? Or a facility with statistics? Or maybe the ability to communicate crisply and clearly? Many would argue that at the very top of the list comes problem solving: that is, the ability to think through and come up with an optimal course of action to address any complex challenge—in business, in public policy, or indeed in life.

Looked at this way, it’s no surprise that McKinsey takes problem solving very seriously, testing for it during the recruiting process and then honing it, in McKinsey consultants, through immersion in a structured seven-step method. To discuss the art of problem solving, I sat down in California with McKinsey senior partner Hugo Sarrazin and also with Charles Conn. Charles is a former McKinsey partner, entrepreneur, executive, and coauthor of the book Bulletproof Problem Solving: The One Skill That Changes Everything [John Wiley & Sons, 2018].

Charles and Hugo, welcome to the podcast. Thank you for being here.

Hugo Sarrazin: Our pleasure.

Charles Conn: It’s terrific to be here.

Simon London: Problem solving is a really interesting piece of terminology. It could mean so many different things. I have a son who’s a teenage climber. They talk about solving problems. Climbing is problem solving. Charles, when you talk about problem solving, what are you talking about?

Charles Conn: For me, problem solving is the answer to the question “What should I do?” It’s interesting when there’s uncertainty and complexity, and when it’s meaningful because there are consequences. Your son’s climbing is a perfect example. There are consequences, and it’s complicated, and there’s uncertainty—can he make that grab? I think we can apply that same frame almost at any level. You can think about questions like “What town would I like to live in?” or “Should I put solar panels on my roof?”

You might think that’s a funny thing to apply problem solving to, but in my mind it’s not fundamentally different from business problem solving, which answers the question “What should my strategy be?” Or problem solving at the policy level: “How do we combat climate change?” “Should I support the local school bond?” I think these are all part and parcel of the same type of question, “What should I do?”

I’m a big fan of structured problem solving. By following steps, we can more clearly understand what problem it is we’re solving, what are the components of the problem that we’re solving, which components are the most important ones for us to pay attention to, which analytic techniques we should apply to those, and how we can synthesize what we’ve learned back into a compelling story. That’s all it is, at its heart.

I think sometimes when people think about seven steps, they assume that there’s a rigidity to this. That’s not it at all. It’s actually to give you the scope for creativity, which often doesn’t exist when your problem solving is muddled.

Simon London: You were just talking about the seven-step process. That’s what’s written down in the book, but it’s a very McKinsey process as well. Without getting too deep into the weeds, let’s go through the steps, one by one. You were just talking about problem definition as being a particularly important thing to get right first. That’s the first step. Hugo, tell us about that.

Hugo Sarrazin: It is surprising how often people jump past this step and make a bunch of assumptions. The most powerful thing is to step back and ask the basic questions—“What are we trying to solve? What are the constraints that exist? What are the dependencies?” Let’s make those explicit and really push the thinking and defining. At McKinsey, we spend an enormous amount of time in writing that little statement, and the statement, if you’re a logic purist, is great. You debate. “Is it an ‘or’? Is it an ‘and’? What’s the action verb?” Because all these specific words help you get to the heart of what matters.

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Simon London: So this is a concise problem statement.

Hugo Sarrazin: Yeah. It’s not like “Can we grow in Japan?” That’s interesting, but it is “What, specifically, are we trying to uncover in the growth of a product in Japan? Or a segment in Japan? Or a channel in Japan?” When you spend an enormous amount of time, in the first meeting of the different stakeholders, debating this and having different people put forward what they think the problem definition is, you realize that people have completely different views of why they’re here. That, to me, is the most important step.

Charles Conn: I would agree with that. For me, the problem context is critical. When we understand “What are the forces acting upon your decision maker? How quickly is the answer needed? With what precision is the answer needed? Are there areas that are off limits or areas where we would particularly like to find our solution? Is the decision maker open to exploring other areas?” then you not only become more efficient, and move toward what we call the critical path in problem solving, but you also make it so much more likely that you’re not going to waste your time or your decision maker’s time.

How often do especially bright young people run off with half of the idea about what the problem is and start collecting data and start building models—only to discover that they’ve really gone off half-cocked.

Hugo Sarrazin: Yeah.

Charles Conn: And in the wrong direction.

Simon London: OK. So step one—and there is a real art and a structure to it—is define the problem. Step two, Charles?

Charles Conn: My favorite step is step two, which is to use logic trees to disaggregate the problem. Every problem we’re solving has some complexity and some uncertainty in it. The only way that we can really get our team working on the problem is to take the problem apart into logical pieces.

What we find, of course, is that the way to disaggregate the problem often gives you an insight into the answer to the problem quite quickly. I love to do two or three different cuts at it, each one giving a bit of a different insight into what might be going wrong. By doing sensible disaggregations, using logic trees, we can figure out which parts of the problem we should be looking at, and we can assign those different parts to team members.

Simon London: What’s a good example of a logic tree on a sort of ratable problem?

Charles Conn: Maybe the easiest one is the classic profit tree. Almost in every business that I would take a look at, I would start with a profit or return-on-assets tree. In its simplest form, you have the components of revenue, which are price and quantity, and the components of cost, which are cost and quantity. Each of those can be broken out. Cost can be broken into variable cost and fixed cost. The components of price can be broken into what your pricing scheme is. That simple tree often provides insight into what’s going on in a business or what the difference is between that business and the competitors.

If we add the leg, which is “What’s the asset base or investment element?”—so profit divided by assets—then we can ask the question “Is the business using its investments sensibly?” whether that’s in stores or in manufacturing or in transportation assets. I hope we can see just how simple this is, even though we’re describing it in words.

When I went to work with Gordon Moore at the Moore Foundation, the problem that he asked us to look at was “How can we save Pacific salmon?” Now, that sounds like an impossible question, but it was amenable to precisely the same type of disaggregation and allowed us to organize what became a 15-year effort to improve the likelihood of good outcomes for Pacific salmon.

Simon London: Now, is there a danger that your logic tree can be impossibly large? This, I think, brings us onto the third step in the process, which is that you have to prioritize.

Charles Conn: Absolutely. The third step, which we also emphasize, along with good problem definition, is rigorous prioritization—we ask the questions “How important is this lever or this branch of the tree in the overall outcome that we seek to achieve? How much can I move that lever?” Obviously, we try and focus our efforts on ones that have a big impact on the problem and the ones that we have the ability to change. With salmon, ocean conditions turned out to be a big lever, but not one that we could adjust. We focused our attention on fish habitats and fish-harvesting practices, which were big levers that we could affect.

People spend a lot of time arguing about branches that are either not important or that none of us can change. We see it in the public square. When we deal with questions at the policy level—“Should you support the death penalty?” “How do we affect climate change?” “How can we uncover the causes and address homelessness?”—it’s even more important that we’re focusing on levers that are big and movable.

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Simon London: Let’s move swiftly on to step four. You’ve defined your problem, you disaggregate it, you prioritize where you want to analyze—what you want to really look at hard. Then you got to the work plan. Now, what does that mean in practice?

Hugo Sarrazin: Depending on what you’ve prioritized, there are many things you could do. It could be breaking the work among the team members so that people have a clear piece of the work to do. It could be defining the specific analyses that need to get done and executed, and being clear on time lines. There’s always a level-one answer, there’s a level-two answer, there’s a level-three answer. Without being too flippant, I can solve any problem during a good dinner with wine. It won’t have a whole lot of backing.

Simon London: Not going to have a lot of depth to it.

Hugo Sarrazin: No, but it may be useful as a starting point. If the stakes are not that high, that could be OK. If it’s really high stakes, you may need level three and have the whole model validated in three different ways. You need to find a work plan that reflects the level of precision, the time frame you have, and the stakeholders you need to bring along in the exercise.

Charles Conn: I love the way you’ve described that, because, again, some people think of problem solving as a linear thing, but of course what’s critical is that it’s iterative. As you say, you can solve the problem in one day or even one hour.

Charles Conn: We encourage our teams everywhere to do that. We call it the one-day answer or the one-hour answer. In work planning, we’re always iterating. Every time you see a 50-page work plan that stretches out to three months, you know it’s wrong. It will be outmoded very quickly by that learning process that you described. Iterative problem solving is a critical part of this. Sometimes, people think work planning sounds dull, but it isn’t. It’s how we know what’s expected of us and when we need to deliver it and how we’re progressing toward the answer. It’s also the place where we can deal with biases. Bias is a feature of every human decision-making process. If we design our team interactions intelligently, we can avoid the worst sort of biases.

Simon London: Here we’re talking about cognitive biases primarily, right? It’s not that I’m biased against you because of your accent or something. These are the cognitive biases that behavioral sciences have shown we all carry around, things like anchoring, overoptimism—these kinds of things.

Both: Yeah.

Charles Conn: Availability bias is the one that I’m always alert to. You think you’ve seen the problem before, and therefore what’s available is your previous conception of it—and we have to be most careful about that. In any human setting, we also have to be careful about biases that are based on hierarchies, sometimes called sunflower bias. I’m sure, Hugo, with your teams, you make sure that the youngest team members speak first. Not the oldest team members, because it’s easy for people to look at who’s senior and alter their own creative approaches.

Hugo Sarrazin: It’s helpful, at that moment—if someone is asserting a point of view—to ask the question “This was true in what context?” You’re trying to apply something that worked in one context to a different one. That can be deadly if the context has changed, and that’s why organizations struggle to change. You promote all these people because they did something that worked well in the past, and then there’s a disruption in the industry, and they keep doing what got them promoted even though the context has changed.

Simon London: Right. Right.

Hugo Sarrazin: So it’s the same thing in problem solving.

Charles Conn: And it’s why diversity in our teams is so important. It’s one of the best things about the world that we’re in now. We’re likely to have people from different socioeconomic, ethnic, and national backgrounds, each of whom sees problems from a slightly different perspective. It is therefore much more likely that the team will uncover a truly creative and clever approach to problem solving.

Simon London: Let’s move on to step five. You’ve done your work plan. Now you’ve actually got to do the analysis. The thing that strikes me here is that the range of tools that we have at our disposal now, of course, is just huge, particularly with advances in computation, advanced analytics. There’s so many things that you can apply here. Just talk about the analysis stage. How do you pick the right tools?

Charles Conn: For me, the most important thing is that we start with simple heuristics and explanatory statistics before we go off and use the big-gun tools. We need to understand the shape and scope of our problem before we start applying these massive and complex analytical approaches.

Simon London: Would you agree with that?

Hugo Sarrazin: I agree. I think there are so many wonderful heuristics. You need to start there before you go deep into the modeling exercise. There’s an interesting dynamic that’s happening, though. In some cases, for some types of problems, it is even better to set yourself up to maximize your learning. Your problem-solving methodology is test and learn, test and learn, test and learn, and iterate. That is a heuristic in itself, the A/B testing that is used in many parts of the world. So that’s a problem-solving methodology. It’s nothing different. It just uses technology and feedback loops in a fast way. The other one is exploratory data analysis. When you’re dealing with a large-scale problem, and there’s so much data, I can get to the heuristics that Charles was talking about through very clever visualization of data.

You test with your data. You need to set up an environment to do so, but don’t get caught up in neural-network modeling immediately. You’re testing, you’re checking—“Is the data right? Is it sound? Does it make sense?”—before you launch too far.

Simon London: You do hear these ideas—that if you have a big enough data set and enough algorithms, they’re going to find things that you just wouldn’t have spotted, find solutions that maybe you wouldn’t have thought of. Does machine learning sort of revolutionize the problem-solving process? Or are these actually just other tools in the toolbox for structured problem solving?

Charles Conn: It can be revolutionary. There are some areas in which the pattern recognition of large data sets and good algorithms can help us see things that we otherwise couldn’t see. But I do think it’s terribly important we don’t think that this particular technique is a substitute for superb problem solving, starting with good problem definition. Many people use machine learning without understanding algorithms that themselves can have biases built into them. Just as 20 years ago, when we were doing statistical analysis, we knew that we needed good model definition, we still need a good understanding of our algorithms and really good problem definition before we launch off into big data sets and unknown algorithms.

Simon London: Step six. You’ve done your analysis.

Charles Conn: I take six and seven together, and this is the place where young problem solvers often make a mistake. They’ve got their analysis, and they assume that’s the answer, and of course it isn’t the answer. The ability to synthesize the pieces that came out of the analysis and begin to weave those into a story that helps people answer the question “What should I do?” This is back to where we started. If we can’t synthesize, and we can’t tell a story, then our decision maker can’t find the answer to “What should I do?”

Simon London: But, again, these final steps are about motivating people to action, right?

Charles Conn: Yeah.

Simon London: I am slightly torn about the nomenclature of problem solving because it’s on paper, right? Until you motivate people to action, you actually haven’t solved anything.

Charles Conn: I love this question because I think decision-making theory, without a bias to action, is a waste of time. Everything in how I approach this is to help people take action that makes the world better.

Simon London: Hence, these are absolutely critical steps. If you don’t do this well, you’ve just got a bunch of analysis.

Charles Conn: We end up in exactly the same place where we started, which is people speaking across each other, past each other in the public square, rather than actually working together, shoulder to shoulder, to crack these important problems.

Simon London: In the real world, we have a lot of uncertainty—arguably, increasing uncertainty. How do good problem solvers deal with that?

Hugo Sarrazin: At every step of the process. In the problem definition, when you’re defining the context, you need to understand those sources of uncertainty and whether they’re important or not important. It becomes important in the definition of the tree.

You need to think carefully about the branches of the tree that are more certain and less certain as you define them. They don’t have equal weight just because they’ve got equal space on the page. Then, when you’re prioritizing, your prioritization approach may put more emphasis on things that have low probability but huge impact—or, vice versa, may put a lot of priority on things that are very likely and, hopefully, have a reasonable impact. You can introduce that along the way. When you come back to the synthesis, you just need to be nuanced about what you’re understanding, the likelihood.

Often, people lack humility in the way they make their recommendations: “This is the answer.” They’re very precise, and I think we would all be well-served to say, “This is a likely answer under the following sets of conditions” and then make the level of uncertainty clearer, if that is appropriate. It doesn’t mean you’re always in the gray zone; it doesn’t mean you don’t have a point of view. It just means that you can be explicit about the certainty of your answer when you make that recommendation.

Simon London: So it sounds like there is an underlying principle: “Acknowledge and embrace the uncertainty. Don’t pretend that it isn’t there. Be very clear about what the uncertainties are up front, and then build that into every step of the process.”

Hugo Sarrazin: Every step of the process.

Simon London: Yeah. We have just walked through a particular structured methodology for problem solving. But, of course, this is not the only structured methodology for problem solving. One that is also very well-known is design thinking, which comes at things very differently. So, Hugo, I know you have worked with a lot of designers. Just give us a very quick summary. Design thinking—what is it, and how does it relate?

Hugo Sarrazin: It starts with an incredible amount of empathy for the user and uses that to define the problem. It does pause and go out in the wild and spend an enormous amount of time seeing how people interact with objects, seeing the experience they’re getting, seeing the pain points or joy—and uses that to infer and define the problem.

Simon London: Problem definition, but out in the world.

Hugo Sarrazin: With an enormous amount of empathy. There’s a huge emphasis on empathy. Traditional, more classic problem solving is you define the problem based on an understanding of the situation. This one almost presupposes that we don’t know the problem until we go see it. The second thing is you need to come up with multiple scenarios or answers or ideas or concepts, and there’s a lot of divergent thinking initially. That’s slightly different, versus the prioritization, but not for long. Eventually, you need to kind of say, “OK, I’m going to converge again.” Then you go and you bring things back to the customer and get feedback and iterate. Then you rinse and repeat, rinse and repeat. There’s a lot of tactile building, along the way, of prototypes and things like that. It’s very iterative.

Simon London: So, Charles, are these complements or are these alternatives?

Charles Conn: I think they’re entirely complementary, and I think Hugo’s description is perfect. When we do problem definition well in classic problem solving, we are demonstrating the kind of empathy, at the very beginning of our problem, that design thinking asks us to approach. When we ideate—and that’s very similar to the disaggregation, prioritization, and work-planning steps—we do precisely the same thing, and often we use contrasting teams, so that we do have divergent thinking. The best teams allow divergent thinking to bump them off whatever their initial biases in problem solving are. For me, design thinking gives us a constant reminder of creativity, empathy, and the tactile nature of problem solving, but it’s absolutely complementary, not alternative.

Simon London: I think, in a world of cross-functional teams, an interesting question is do people with design-thinking backgrounds really work well together with classical problem solvers? How do you make that chemistry happen?

Hugo Sarrazin: Yeah, it is not easy when people have spent an enormous amount of time seeped in design thinking or user-centric design, whichever word you want to use. If the person who’s applying classic problem-solving methodology is very rigid and mechanical in the way they’re doing it, there could be an enormous amount of tension. If there’s not clarity in the role and not clarity in the process, I think having the two together can be, sometimes, problematic.

The second thing that happens often is that the artifacts the two methodologies try to gravitate toward can be different. Classic problem solving often gravitates toward a model; design thinking migrates toward a prototype. Rather than writing a big deck with all my supporting evidence, they’ll bring an example, a thing, and that feels different. Then you spend your time differently to achieve those two end products, so that’s another source of friction.

Now, I still think it can be an incredibly powerful thing to have the two—if there are the right people with the right mind-set, if there is a team that is explicit about the roles, if we’re clear about the kind of outcomes we are attempting to bring forward. There’s an enormous amount of collaborativeness and respect.

Simon London: But they have to respect each other’s methodology and be prepared to flex, maybe, a little bit, in how this process is going to work.

Hugo Sarrazin: Absolutely.

Simon London: The other area where, it strikes me, there could be a little bit of a different sort of friction is this whole concept of the day-one answer, which is what we were just talking about in classical problem solving. Now, you know that this is probably not going to be your final answer, but that’s how you begin to structure the problem. Whereas I would imagine your design thinkers—no, they’re going off to do their ethnographic research and get out into the field, potentially for a long time, before they come back with at least an initial hypothesis.

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Hugo Sarrazin: That is a great callout, and that’s another difference. Designers typically will like to soak into the situation and avoid converging too quickly. There’s optionality and exploring different options. There’s a strong belief that keeps the solution space wide enough that you can come up with more radical ideas. If there’s a large design team or many designers on the team, and you come on Friday and say, “What’s our week-one answer?” they’re going to struggle. They’re not going to be comfortable, naturally, to give that answer. It doesn’t mean they don’t have an answer; it’s just not where they are in their thinking process.

Simon London: I think we are, sadly, out of time for today. But Charles and Hugo, thank you so much.

Charles Conn: It was a pleasure to be here, Simon.

Hugo Sarrazin: It was a pleasure. Thank you.

Simon London: And thanks, as always, to you, our listeners, for tuning into this episode of the McKinsey Podcast . If you want to learn more about problem solving, you can find the book, Bulletproof Problem Solving: The One Skill That Changes Everything , online or order it through your local bookstore. To learn more about McKinsey, you can of course find us at McKinsey.com.

Charles Conn is CEO of Oxford Sciences Innovation and an alumnus of McKinsey’s Sydney office. Hugo Sarrazin is a senior partner in the Silicon Valley office, where Simon London, a member of McKinsey Publishing, is also based.

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Strategy to beat the odds

7 Module 7: Thinking, Reasoning, and Problem-Solving

This module is about how a solid working knowledge of psychological principles can help you to think more effectively, so you can succeed in school and life. You might be inclined to believe that—because you have been thinking for as long as you can remember, because you are able to figure out the solution to many problems, because you feel capable of using logic to argue a point, because you can evaluate whether the things you read and hear make sense—you do not need any special training in thinking. But this, of course, is one of the key barriers to helping people think better. If you do not believe that there is anything wrong, why try to fix it?

The human brain is indeed a remarkable thinking machine, capable of amazing, complex, creative, logical thoughts. Why, then, are we telling you that you need to learn how to think? Mainly because one major lesson from cognitive psychology is that these capabilities of the human brain are relatively infrequently realized. Many psychologists believe that people are essentially “cognitive misers.” It is not that we are lazy, but that we have a tendency to expend the least amount of mental effort necessary. Although you may not realize it, it actually takes a great deal of energy to think. Careful, deliberative reasoning and critical thinking are very difficult. Because we seem to be successful without going to the trouble of using these skills well, it feels unnecessary to develop them. As you shall see, however, there are many pitfalls in the cognitive processes described in this module. When people do not devote extra effort to learning and improving reasoning, problem solving, and critical thinking skills, they make many errors.

As is true for memory, if you develop the cognitive skills presented in this module, you will be more successful in school. It is important that you realize, however, that these skills will help you far beyond school, even more so than a good memory will. Although it is somewhat useful to have a good memory, ten years from now no potential employer will care how many questions you got right on multiple choice exams during college. All of them will, however, recognize whether you are a logical, analytical, critical thinker. With these thinking skills, you will be an effective, persuasive communicator and an excellent problem solver.

The module begins by describing different kinds of thought and knowledge, especially conceptual knowledge and critical thinking. An understanding of these differences will be valuable as you progress through school and encounter different assignments that require you to tap into different kinds of knowledge. The second section covers deductive and inductive reasoning, which are processes we use to construct and evaluate strong arguments. They are essential skills to have whenever you are trying to persuade someone (including yourself) of some point, or to respond to someone’s efforts to persuade you. The module ends with a section about problem solving. A solid understanding of the key processes involved in problem solving will help you to handle many daily challenges.

7.1. Different kinds of thought

7.2. Reasoning and Judgment

7.3. Problem Solving

READING WITH PURPOSE

Remember and understand.

By reading and studying Module 7, you should be able to remember and describe:

• Concepts and inferences (7.1)
• Procedural knowledge (7.1)
• Metacognition (7.1)
• Characteristics of critical thinking:  skepticism; identify biases, distortions, omissions, and assumptions; reasoning and problem solving skills  (7.1)
• Reasoning:  deductive reasoning, deductively valid argument, inductive reasoning, inductively strong argument, availability heuristic, representativeness heuristic  (7.2)
• Fixation:  functional fixedness, mental set  (7.3)
• Algorithms, heuristics, and the role of confirmation bias (7.3)
• Effective problem solving sequence (7.3)

By reading and thinking about how the concepts in Module 6 apply to real life, you should be able to:

• Identify which type of knowledge a piece of information is (7.1)
• Recognize examples of deductive and inductive reasoning (7.2)
• Recognize judgments that have probably been influenced by the availability heuristic (7.2)
• Recognize examples of problem solving heuristics and algorithms (7.3)

Analyze, Evaluate, and Create

By reading and thinking about Module 6, participating in classroom activities, and completing out-of-class assignments, you should be able to:

• Use the principles of critical thinking to evaluate information (7.1)
• Explain whether examples of reasoning arguments are deductively valid or inductively strong (7.2)
• Outline how you could try to solve a problem from your life using the effective problem solving sequence (7.3)

7.1. Different kinds of thought and knowledge

• Take a few minutes to write down everything that you know about dogs.
• Do you believe that:
• Psychic ability exists?
• Hypnosis is an altered state of consciousness?
• Magnet therapy is effective for relieving pain?
• Aerobic exercise is an effective treatment for depression?
• UFO’s from outer space have visited earth?

On what do you base your belief or disbelief for the questions above?

Of course, we all know what is meant by the words  think  and  knowledge . You probably also realize that they are not unitary concepts; there are different kinds of thought and knowledge. In this section, let us look at some of these differences. If you are familiar with these different kinds of thought and pay attention to them in your classes, it will help you to focus on the right goals, learn more effectively, and succeed in school. Different assignments and requirements in school call on you to use different kinds of knowledge or thought, so it will be very helpful for you to learn to recognize them (Anderson, et al. 2001).

Factual and conceptual knowledge

Module 5 introduced the idea of declarative memory, which is composed of facts and episodes. If you have ever played a trivia game or watched Jeopardy on TV, you realize that the human brain is able to hold an extraordinary number of facts. Likewise, you realize that each of us has an enormous store of episodes, essentially facts about events that happened in our own lives. It may be difficult to keep that in mind when we are struggling to retrieve one of those facts while taking an exam, however. Part of the problem is that, in contradiction to the advice from Module 5, many students continue to try to memorize course material as a series of unrelated facts (picture a history student simply trying to memorize history as a set of unrelated dates without any coherent story tying them together). Facts in the real world are not random and unorganized, however. It is the way that they are organized that constitutes a second key kind of knowledge, conceptual.

Concepts are nothing more than our mental representations of categories of things in the world. For example, think about dogs. When you do this, you might remember specific facts about dogs, such as they have fur and they bark. You may also recall dogs that you have encountered and picture them in your mind. All of this information (and more) makes up your concept of dog. You can have concepts of simple categories (e.g., triangle), complex categories (e.g., small dogs that sleep all day, eat out of the garbage, and bark at leaves), kinds of people (e.g., psychology professors), events (e.g., birthday parties), and abstract ideas (e.g., justice). Gregory Murphy (2002) refers to concepts as the “glue that holds our mental life together” (p. 1). Very simply, summarizing the world by using concepts is one of the most important cognitive tasks that we do. Our conceptual knowledge  is  our knowledge about the world. Individual concepts are related to each other to form a rich interconnected network of knowledge. For example, think about how the following concepts might be related to each other: dog, pet, play, Frisbee, chew toy, shoe. Or, of more obvious use to you now, how these concepts are related: working memory, long-term memory, declarative memory, procedural memory, and rehearsal? Because our minds have a natural tendency to organize information conceptually, when students try to remember course material as isolated facts, they are working against their strengths.

One last important point about concepts is that they allow you to instantly know a great deal of information about something. For example, if someone hands you a small red object and says, “here is an apple,” they do not have to tell you, “it is something you can eat.” You already know that you can eat it because it is true by virtue of the fact that the object is an apple; this is called drawing an  inference , assuming that something is true on the basis of your previous knowledge (for example, of category membership or of how the world works) or logical reasoning.

Procedural knowledge

Physical skills, such as tying your shoes, doing a cartwheel, and driving a car (or doing all three at the same time, but don’t try this at home) are certainly a kind of knowledge. They are procedural knowledge, the same idea as procedural memory that you saw in Module 5. Mental skills, such as reading, debating, and planning a psychology experiment, are procedural knowledge, as well. In short, procedural knowledge is the knowledge how to do something (Cohen & Eichenbaum, 1993).

Metacognitive knowledge

Floyd used to think that he had a great memory. Now, he has a better memory. Why? Because he finally realized that his memory was not as great as he once thought it was. Because Floyd eventually learned that he often forgets where he put things, he finally developed the habit of putting things in the same place. (Unfortunately, he did not learn this lesson before losing at least 5 watches and a wedding ring.) Because he finally realized that he often forgets to do things, he finally started using the To Do list app on his phone. And so on. Floyd’s insights about the real limitations of his memory have allowed him to remember things that he used to forget.

All of us have knowledge about the way our own minds work. You may know that you have a good memory for people’s names and a poor memory for math formulas. Someone else might realize that they have difficulty remembering to do things, like stopping at the store on the way home. Others still know that they tend to overlook details. This knowledge about our own thinking is actually quite important; it is called metacognitive knowledge, or  metacognition . Like other kinds of thinking skills, it is subject to error. For example, in unpublished research, one of the authors surveyed about 120 General Psychology students on the first day of the term. Among other questions, the students were asked them to predict their grade in the class and report their current Grade Point Average. Two-thirds of the students predicted that their grade in the course would be higher than their GPA. (The reality is that at our college, students tend to earn lower grades in psychology than their overall GPA.) Another example: Students routinely report that they thought they had done well on an exam, only to discover, to their dismay, that they were wrong (more on that important problem in a moment). Both errors reveal a breakdown in metacognition.

The Dunning-Kruger Effect

In general, most college students probably do not study enough. For example, using data from the National Survey of Student Engagement, Fosnacht, McCormack, and Lerma (2018) reported that first-year students at 4-year colleges in the U.S. averaged less than 14 hours per week preparing for classes. The typical suggestion is that you should spend two hours outside of class for every hour in class, or 24 – 30 hours per week for a full-time student. Clearly, students in general are nowhere near that recommended mark. Many observers, including some faculty, believe that this shortfall is a result of students being too busy or lazy. Now, it may be true that many students are too busy, with work and family obligations, for example. Others, are not particularly motivated in school, and therefore might correctly be labeled lazy. A third possible explanation, however, is that some students might not think they need to spend this much time. And this is a matter of metacognition. Consider the scenario that we mentioned above, students thinking they had done well on an exam only to discover that they did not. Justin Kruger and David Dunning examined scenarios very much like this in 1999. Kruger and Dunning gave research participants tests measuring humor, logic, and grammar. Then, they asked the participants to assess their own abilities and test performance in these areas. They found that participants in general tended to overestimate their abilities, already a problem with metacognition. Importantly, the participants who scored the lowest overestimated their abilities the most. Specifically, students who scored in the bottom quarter (averaging in the 12th percentile) thought they had scored in the 62nd percentile. This has become known as the  Dunning-Kruger effect . Many individual faculty members have replicated these results with their own student on their course exams, including the authors of this book. Think about it. Some students who just took an exam and performed poorly believe that they did well before seeing their score. It seems very likely that these are the very same students who stopped studying the night before because they thought they were “done.” Quite simply, it is not just that they did not know the material. They did not know that they did not know the material. That is poor metacognition.

In order to develop good metacognitive skills, you should continually monitor your thinking and seek frequent feedback on the accuracy of your thinking (Medina, Castleberry, & Persky 2017). For example, in classes get in the habit of predicting your exam grades. As soon as possible after taking an exam, try to find out which questions you missed and try to figure out why. If you do this soon enough, you may be able to recall the way it felt when you originally answered the question. Did you feel confident that you had answered the question correctly? Then you have just discovered an opportunity to improve your metacognition. Be on the lookout for that feeling and respond with caution.

concept :  a mental representation of a category of things in the world

Dunning-Kruger effect : individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

inference : an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

metacognition :  knowledge about one’s own cognitive processes; thinking about your thinking

Critical thinking

One particular kind of knowledge or thinking skill that is related to metacognition is  critical thinking (Chew, 2020). You may have noticed that critical thinking is an objective in many college courses, and thus it could be a legitimate topic to cover in nearly any college course. It is particularly appropriate in psychology, however. As the science of (behavior and) mental processes, psychology is obviously well suited to be the discipline through which you should be introduced to this important way of thinking.

More importantly, there is a particular need to use critical thinking in psychology. We are all, in a way, experts in human behavior and mental processes, having engaged in them literally since birth. Thus, perhaps more than in any other class, students typically approach psychology with very clear ideas and opinions about its subject matter. That is, students already “know” a lot about psychology. The problem is, “it ain’t so much the things we don’t know that get us into trouble. It’s the things we know that just ain’t so” (Ward, quoted in Gilovich 1991). Indeed, many of students’ preconceptions about psychology are just plain wrong. Randolph Smith (2002) wrote a book about critical thinking in psychology called  Challenging Your Preconceptions,  highlighting this fact. On the other hand, many of students’ preconceptions about psychology are just plain right! But wait, how do you know which of your preconceptions are right and which are wrong? And when you come across a research finding or theory in this class that contradicts your preconceptions, what will you do? Will you stick to your original idea, discounting the information from the class? Will you immediately change your mind? Critical thinking can help us sort through this confusing mess.

But what is critical thinking? The goal of critical thinking is simple to state (but extraordinarily difficult to achieve): it is to be right, to draw the correct conclusions, to believe in things that are true and to disbelieve things that are false. We will provide two definitions of critical thinking (or, if you like, one large definition with two distinct parts). First, a more conceptual one: Critical thinking is thinking like a scientist in your everyday life (Schmaltz, Jansen, & Wenckowski, 2017).  Our second definition is more operational; it is simply a list of skills that are essential to be a critical thinker. Critical thinking entails solid reasoning and problem solving skills; skepticism; and an ability to identify biases, distortions, omissions, and assumptions. Excellent deductive and inductive reasoning, and problem solving skills contribute to critical thinking. So, you can consider the subject matter of sections 7.2 and 7.3 to be part of critical thinking. Because we will be devoting considerable time to these concepts in the rest of the module, let us begin with a discussion about the other aspects of critical thinking.

Let’s address that first part of the definition. Scientists form hypotheses, or predictions about some possible future observations. Then, they collect data, or information (think of this as making those future observations). They do their best to make unbiased observations using reliable techniques that have been verified by others. Then, and only then, they draw a conclusion about what those observations mean. Oh, and do not forget the most important part. “Conclusion” is probably not the most appropriate word because this conclusion is only tentative. A scientist is always prepared that someone else might come along and produce new observations that would require a new conclusion be drawn. Wow! If you like to be right, you could do a lot worse than using a process like this.

A Critical Thinker’s Toolkit 

Now for the second part of the definition. Good critical thinkers (and scientists) rely on a variety of tools to evaluate information. Perhaps the most recognizable tool for critical thinking is  skepticism (and this term provides the clearest link to the thinking like a scientist definition, as you are about to see). Some people intend it as an insult when they call someone a skeptic. But if someone calls you a skeptic, if they are using the term correctly, you should consider it a great compliment. Simply put, skepticism is a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided. People from Missouri should recognize this principle, as Missouri is known as the Show-Me State. As a skeptic, you are not inclined to believe something just because someone said so, because someone else believes it, or because it sounds reasonable. You must be persuaded by high quality evidence.

Of course, if that evidence is produced, you have a responsibility as a skeptic to change your belief. Failure to change a belief in the face of good evidence is not skepticism; skepticism has open mindedness at its core. M. Neil Browne and Stuart Keeley (2018) use the term weak sense critical thinking to describe critical thinking behaviors that are used only to strengthen a prior belief. Strong sense critical thinking, on the other hand, has as its goal reaching the best conclusion. Sometimes that means strengthening your prior belief, but sometimes it means changing your belief to accommodate the better evidence.

Many times, a failure to think critically or weak sense critical thinking is related to a  bias , an inclination, tendency, leaning, or prejudice. Everybody has biases, but many people are unaware of them. Awareness of your own biases gives you the opportunity to control or counteract them. Unfortunately, however, many people are happy to let their biases creep into their attempts to persuade others; indeed, it is a key part of their persuasive strategy. To see how these biases influence messages, just look at the different descriptions and explanations of the same events given by people of different ages or income brackets, or conservative versus liberal commentators, or by commentators from different parts of the world. Of course, to be successful, these people who are consciously using their biases must disguise them. Even undisguised biases can be difficult to identify, so disguised ones can be nearly impossible.

Here are some common sources of biases:

• Personal values and beliefs.  Some people believe that human beings are basically driven to seek power and that they are typically in competition with one another over scarce resources. These beliefs are similar to the world-view that political scientists call “realism.” Other people believe that human beings prefer to cooperate and that, given the chance, they will do so. These beliefs are similar to the world-view known as “idealism.” For many people, these deeply held beliefs can influence, or bias, their interpretations of such wide ranging situations as the behavior of nations and their leaders or the behavior of the driver in the car ahead of you. For example, if your worldview is that people are typically in competition and someone cuts you off on the highway, you may assume that the driver did it purposely to get ahead of you. Other types of beliefs about the way the world is or the way the world should be, for example, political beliefs, can similarly become a significant source of bias.
• Racism, sexism, ageism and other forms of prejudice and bigotry.  These are, sadly, a common source of bias in many people. They are essentially a special kind of “belief about the way the world is.” These beliefs—for example, that women do not make effective leaders—lead people to ignore contradictory evidence (examples of effective women leaders, or research that disputes the belief) and to interpret ambiguous evidence in a way consistent with the belief.
• Self-interest.  When particular people benefit from things turning out a certain way, they can sometimes be very susceptible to letting that interest bias them. For example, a company that will earn a profit if they sell their product may have a bias in the way that they give information about their product. A union that will benefit if its members get a generous contract might have a bias in the way it presents information about salaries at competing organizations. (Note that our inclusion of examples describing both companies and unions is an explicit attempt to control for our own personal biases). Home buyers are often dismayed to discover that they purchased their dream house from someone whose self-interest led them to lie about flooding problems in the basement or back yard. This principle, the biasing power of self-interest, is likely what led to the famous phrase  Caveat Emptor  (let the buyer beware) .  

Knowing that these types of biases exist will help you evaluate evidence more critically. Do not forget, though, that people are not always keen to let you discover the sources of biases in their arguments. For example, companies or political organizations can sometimes disguise their support of a research study by contracting with a university professor, who comes complete with a seemingly unbiased institutional affiliation, to conduct the study.

People’s biases, conscious or unconscious, can lead them to make omissions, distortions, and assumptions that undermine our ability to correctly evaluate evidence. It is essential that you look for these elements. Always ask, what is missing, what is not as it appears, and what is being assumed here? For example, consider this (fictional) chart from an ad reporting customer satisfaction at 4 local health clubs.

Clearly, from the results of the chart, one would be tempted to give Club C a try, as customer satisfaction is much higher than for the other 3 clubs.

There are so many distortions and omissions in this chart, however, that it is actually quite meaningless. First, how was satisfaction measured? Do the bars represent responses to a survey? If so, how were the questions asked? Most importantly, where is the missing scale for the chart? Although the differences look quite large, are they really?

Well, here is the same chart, with a different scale, this time labeled:

Club C is not so impressive any more, is it? In fact, all of the health clubs have customer satisfaction ratings (whatever that means) between 85% and 88%. In the first chart, the entire scale of the graph included only the percentages between 83 and 89. This “judicious” choice of scale—some would call it a distortion—and omission of that scale from the chart make the tiny differences among the clubs seem important, however.

Also, in order to be a critical thinker, you need to learn to pay attention to the assumptions that underlie a message. Let us briefly illustrate the role of assumptions by touching on some people’s beliefs about the criminal justice system in the US. Some believe that a major problem with our judicial system is that many criminals go free because of legal technicalities. Others believe that a major problem is that many innocent people are convicted of crimes. The simple fact is, both types of errors occur. A person’s conclusion about which flaw in our judicial system is the greater tragedy is based on an assumption about which of these is the more serious error (letting the guilty go free or convicting the innocent). This type of assumption is called a value assumption (Browne and Keeley, 2018). It reflects the differences in values that people develop, differences that may lead us to disregard valid evidence that does not fit in with our particular values.

Oh, by the way, some students probably noticed this, but the seven tips for evaluating information that we shared in Module 1 are related to this. Actually, they are part of this section. The tips are, to a very large degree, set of ideas you can use to help you identify biases, distortions, omissions, and assumptions. If you do not remember this section, we strongly recommend you take a few minutes to review it.

skepticism :  a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

bias : an inclination, tendency, leaning, or prejudice

• Which of your beliefs (or disbeliefs) from the Activate exercise for this section were derived from a process of critical thinking? If some of your beliefs were not based on critical thinking, are you willing to reassess these beliefs? If the answer is no, why do you think that is? If the answer is yes, what concrete steps will you take?

7.2 Reasoning and Judgment

• What percentage of kidnappings are committed by strangers?
• Which area of the house is riskiest: kitchen, bathroom, or stairs?
• What is the most common cancer in the US?
• What percentage of workplace homicides are committed by co-workers?

An essential set of procedural thinking skills is  reasoning , the ability to generate and evaluate solid conclusions from a set of statements or evidence. You should note that these conclusions (when they are generated instead of being evaluated) are one key type of inference that we described in Section 7.1. There are two main types of reasoning, deductive and inductive.

Deductive reasoning

Suppose your teacher tells you that if you get an A on the final exam in a course, you will get an A for the whole course. Then, you get an A on the final exam. What will your final course grade be? Most people can see instantly that you can conclude with certainty that you will get an A for the course. This is a type of reasoning called  deductive reasoning , which is defined as reasoning in which a conclusion is guaranteed to be true as long as the statements leading to it are true. The three statements can be listed as an  argument , with two beginning statements and a conclusion:

Statement 1: If you get an A on the final exam, you will get an A for the course

Statement 2: You get an A on the final exam

Conclusion: You will get an A for the course

This particular arrangement, in which true beginning statements lead to a guaranteed true conclusion, is known as a  deductively valid argument . Although deductive reasoning is often the subject of abstract, brain-teasing, puzzle-like word problems, it is actually an extremely important type of everyday reasoning. It is just hard to recognize sometimes. For example, imagine that you are looking for your car keys and you realize that they are either in the kitchen drawer or in your book bag. After looking in the kitchen drawer, you instantly know that they must be in your book bag. That conclusion results from a simple deductive reasoning argument. In addition, solid deductive reasoning skills are necessary for you to succeed in the sciences, philosophy, math, computer programming, and any endeavor involving the use of logic to persuade others to your point of view or to evaluate others’ arguments.

Cognitive psychologists, and before them philosophers, have been quite interested in deductive reasoning, not so much for its practical applications, but for the insights it can offer them about the ways that human beings think. One of the early ideas to emerge from the examination of deductive reasoning is that people learn (or develop) mental versions of rules that allow them to solve these types of reasoning problems (Braine, 1978; Braine, Reiser, & Rumain, 1984). The best way to see this point of view is to realize that there are different possible rules, and some of them are very simple. For example, consider this rule of logic:

therefore q

Logical rules are often presented abstractly, as letters, in order to imply that they can be used in very many specific situations. Here is a concrete version of the of the same rule:

I’ll either have pizza or a hamburger for dinner tonight (p or q)

I won’t have pizza (not p)

Therefore, I’ll have a hamburger (therefore q)

This kind of reasoning seems so natural, so easy, that it is quite plausible that we would use a version of this rule in our daily lives. At least, it seems more plausible than some of the alternative possibilities—for example, that we need to have experience with the specific situation (pizza or hamburger, in this case) in order to solve this type of problem easily. So perhaps there is a form of natural logic (Rips, 1990) that contains very simple versions of logical rules. When we are faced with a reasoning problem that maps onto one of these rules, we use the rule.

But be very careful; things are not always as easy as they seem. Even these simple rules are not so simple. For example, consider the following rule. Many people fail to realize that this rule is just as valid as the pizza or hamburger rule above.

if p, then q

therefore, not p

Concrete version:

If I eat dinner, then I will have dessert

I did not have dessert

Therefore, I did not eat dinner

The simple fact is, it can be very difficult for people to apply rules of deductive logic correctly; as a result, they make many errors when trying to do so. Is this a deductively valid argument or not?

Students who like school study a lot

Students who study a lot get good grades

Jane does not like school

Therefore, Jane does not get good grades

Many people are surprised to discover that this is not a logically valid argument; the conclusion is not guaranteed to be true from the beginning statements. Although the first statement says that students who like school study a lot, it does NOT say that students who do not like school do not study a lot. In other words, it may very well be possible to study a lot without liking school. Even people who sometimes get problems like this right might not be using the rules of deductive reasoning. Instead, they might just be making judgments for examples they know, in this case, remembering instances of people who get good grades despite not liking school.

Making deductive reasoning even more difficult is the fact that there are two important properties that an argument may have. One, it can be valid or invalid (meaning that the conclusion does or does not follow logically from the statements leading up to it). Two, an argument (or more correctly, its conclusion) can be true or false. Here is an example of an argument that is logically valid, but has a false conclusion (at least we think it is false).

Either you are eleven feet tall or the Grand Canyon was created by a spaceship crashing into the earth.

You are not eleven feet tall

Therefore the Grand Canyon was created by a spaceship crashing into the earth

This argument has the exact same form as the pizza or hamburger argument above, making it is deductively valid. The conclusion is so false, however, that it is absurd (of course, the reason the conclusion is false is that the first statement is false). When people are judging arguments, they tend to not observe the difference between deductive validity and the empirical truth of statements or conclusions. If the elements of an argument happen to be true, people are likely to judge the argument logically valid; if the elements are false, they will very likely judge it invalid (Markovits & Bouffard-Bouchard, 1992; Moshman & Franks, 1986). Thus, it seems a stretch to say that people are using these logical rules to judge the validity of arguments. Many psychologists believe that most people actually have very limited deductive reasoning skills (Johnson-Laird, 1999). They argue that when faced with a problem for which deductive logic is required, people resort to some simpler technique, such as matching terms that appear in the statements and the conclusion (Evans, 1982). This might not seem like a problem, but what if reasoners believe that the elements are true and they happen to be wrong; they will would believe that they are using a form of reasoning that guarantees they are correct and yet be wrong.

deductive reasoning :  a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

argument :  a set of statements in which the beginning statements lead to a conclusion

deductively valid argument :  an argument for which true beginning statements guarantee that the conclusion is true

Inductive reasoning and judgment

Every day, you make many judgments about the likelihood of one thing or another. Whether you realize it or not, you are practicing  inductive reasoning   on a daily basis. In inductive reasoning arguments, a conclusion is likely whenever the statements preceding it are true. The first thing to notice about inductive reasoning is that, by definition, you can never be sure about your conclusion; you can only estimate how likely the conclusion is. Inductive reasoning may lead you to focus on Memory Encoding and Recoding when you study for the exam, but it is possible the instructor will ask more questions about Memory Retrieval instead. Unlike deductive reasoning, the conclusions you reach through inductive reasoning are only probable, not certain. That is why scientists consider inductive reasoning weaker than deductive reasoning. But imagine how hard it would be for us to function if we could not act unless we were certain about the outcome.

Inductive reasoning can be represented as logical arguments consisting of statements and a conclusion, just as deductive reasoning can be. In an inductive argument, you are given some statements and a conclusion (or you are given some statements and must draw a conclusion). An argument is  inductively strong   if the conclusion would be very probable whenever the statements are true. So, for example, here is an inductively strong argument:

• Statement #1: The forecaster on Channel 2 said it is going to rain today.
• Statement #2: The forecaster on Channel 5 said it is going to rain today.
• Statement #3: It is very cloudy and humid.
• Statement #4: You just heard thunder.
• Conclusion (or judgment): It is going to rain today.

Think of the statements as evidence, on the basis of which you will draw a conclusion. So, based on the evidence presented in the four statements, it is very likely that it will rain today. Will it definitely rain today? Certainly not. We can all think of times that the weather forecaster was wrong.

A true story: Some years ago psychology student was watching a baseball playoff game between the St. Louis Cardinals and the Los Angeles Dodgers. A graphic on the screen had just informed the audience that the Cardinal at bat, (Hall of Fame shortstop) Ozzie Smith, a switch hitter batting left-handed for this plate appearance, had never, in nearly 3000 career at-bats, hit a home run left-handed. The student, who had just learned about inductive reasoning in his psychology class, turned to his companion (a Cardinals fan) and smugly said, “It is an inductively strong argument that Ozzie Smith will not hit a home run.” He turned back to face the television just in time to watch the ball sail over the right field fence for a home run. Although the student felt foolish at the time, he was not wrong. It was an inductively strong argument; 3000 at-bats is an awful lot of evidence suggesting that the Wizard of Ozz (as he was known) would not be hitting one out of the park (think of each at-bat without a home run as a statement in an inductive argument). Sadly (for the die-hard Cubs fan and Cardinals-hating student), despite the strength of the argument, the conclusion was wrong.

Given the possibility that we might draw an incorrect conclusion even with an inductively strong argument, we really want to be sure that we do, in fact, make inductively strong arguments. If we judge something probable, it had better be probable. If we judge something nearly impossible, it had better not happen. Think of inductive reasoning, then, as making reasonably accurate judgments of the probability of some conclusion given a set of evidence.

We base many decisions in our lives on inductive reasoning. For example:

Statement #1: Psychology is not my best subject

Statement #2: My psychology instructor has a reputation for giving difficult exams

Statement #3: My first psychology exam was much harder than I expected

Judgment: The next exam will probably be very difficult.

Decision: I will study tonight instead of watching Netflix.

Some other examples of judgments that people commonly make in a school context include judgments of the likelihood that:

• A particular class will be interesting/useful/difficult
• You will be able to finish writing a paper by next week if you go out tonight
• Your laptop’s battery will last through the next trip to the library
• You will not miss anything important if you skip class tomorrow
• Your instructor will not notice if you skip class tomorrow
• You will be able to find a book that you will need for a paper
• There will be an essay question about Memory Encoding on the next exam

Tversky and Kahneman (1983) recognized that there are two general ways that we might make these judgments; they termed them extensional (i.e., following the laws of probability) and intuitive (i.e., using shortcuts or heuristics, see below). We will use a similar distinction between Type 1 and Type 2 thinking, as described by Keith Stanovich and his colleagues (Evans and Stanovich, 2013; Stanovich and West, 2000). Type 1 thinking is fast, automatic, effortful, and emotional. In fact, it is hardly fair to call it reasoning at all, as judgments just seem to pop into one’s head. Type 2 thinking , on the other hand, is slow, effortful, and logical. So obviously, it is more likely to lead to a correct judgment, or an optimal decision. The problem is, we tend to over-rely on Type 1. Now, we are not saying that Type 2 is the right way to go for every decision or judgment we make. It seems a bit much, for example, to engage in a step-by-step logical reasoning procedure to decide whether we will have chicken or fish for dinner tonight.

Many bad decisions in some very important contexts, however, can be traced back to poor judgments of the likelihood of certain risks or outcomes that result from the use of Type 1 when a more logical reasoning process would have been more appropriate. For example:

Statement #1: It is late at night.

Statement #2: Albert has been drinking beer for the past five hours at a party.

Statement #3: Albert is not exactly sure where he is or how far away home is.

Judgment: Albert will have no difficulty walking home.

Decision: He walks home alone.

As you can see in this example, the three statements backing up the judgment do not really support it. In other words, this argument is not inductively strong because it is based on judgments that ignore the laws of probability. What are the chances that someone facing these conditions will be able to walk home alone easily? And one need not be drunk to make poor decisions based on judgments that just pop into our heads.

The truth is that many of our probability judgments do not come very close to what the laws of probability say they should be. Think about it. In order for us to reason in accordance with these laws, we would need to know the laws of probability, which would allow us to calculate the relationship between particular pieces of evidence and the probability of some outcome (i.e., how much likelihood should change given a piece of evidence), and we would have to do these heavy math calculations in our heads. After all, that is what Type 2 requires. Needless to say, even if we were motivated, we often do not even know how to apply Type 2 reasoning in many cases.

So what do we do when we don’t have the knowledge, skills, or time required to make the correct mathematical judgment? Do we hold off and wait until we can get better evidence? Do we read up on probability and fire up our calculator app so we can compute the correct probability? Of course not. We rely on Type 1 thinking. We “wing it.” That is, we come up with a likelihood estimate using some means at our disposal. Psychologists use the term heuristic to describe the type of “winging it” we are talking about. A  heuristic   is a shortcut strategy that we use to make some judgment or solve some problem (see Section 7.3). Heuristics are easy and quick, think of them as the basic procedures that are characteristic of Type 1.  They can absolutely lead to reasonably good judgments and decisions in some situations (like choosing between chicken and fish for dinner). They are, however, far from foolproof. There are, in fact, quite a lot of situations in which heuristics can lead us to make incorrect judgments, and in many cases the decisions based on those judgments can have serious consequences.

Let us return to the activity that begins this section. You were asked to judge the likelihood (or frequency) of certain events and risks. You were free to come up with your own evidence (or statements) to make these judgments. This is where a heuristic crops up. As a judgment shortcut, we tend to generate specific examples of those very events to help us decide their likelihood or frequency. For example, if we are asked to judge how common, frequent, or likely a particular type of cancer is, many of our statements would be examples of specific cancer cases:

Statement #1: Andy Kaufman (comedian) had lung cancer.

Statement #2: Colin Powell (US Secretary of State) had prostate cancer.

Statement #3: Bob Marley (musician) had skin and brain cancer

Statement #4: Sandra Day O’Connor (Supreme Court Justice) had breast cancer.

Statement #5: Fred Rogers (children’s entertainer) had stomach cancer.

Statement #6: Robin Roberts (news anchor) had breast cancer.

Statement #7: Bette Davis (actress) had breast cancer.

Judgment: Breast cancer is the most common type.

Your own experience or memory may also tell you that breast cancer is the most common type. But it is not (although it is common). Actually, skin cancer is the most common type in the US. We make the same types of misjudgments all the time because we do not generate the examples or evidence according to their actual frequencies or probabilities. Instead, we have a tendency (or bias) to search for the examples in memory; if they are easy to retrieve, we assume that they are common. To rephrase this in the language of the heuristic, events seem more likely to the extent that they are available to memory. This bias has been termed the  availability heuristic   (Kahneman and Tversky, 1974).

The fact that we use the availability heuristic does not automatically mean that our judgment is wrong. The reason we use heuristics in the first place is that they work fairly well in many cases (and, of course that they are easy to use). So, the easiest examples to think of sometimes are the most common ones. Is it more likely that a member of the U.S. Senate is a man or a woman? Most people have a much easier time generating examples of male senators. And as it turns out, the U.S. Senate has many more men than women (74 to 26 in 2020). In this case, then, the availability heuristic would lead you to make the correct judgment; it is far more likely that a senator would be a man.

In many other cases, however, the availability heuristic will lead us astray. This is because events can be memorable for many reasons other than their frequency. Section 5.2, Encoding Meaning, suggested that one good way to encode the meaning of some information is to form a mental image of it. Thus, information that has been pictured mentally will be more available to memory. Indeed, an event that is vivid and easily pictured will trick many people into supposing that type of event is more common than it actually is. Repetition of information will also make it more memorable. So, if the same event is described to you in a magazine, on the evening news, on a podcast that you listen to, and in your Facebook feed; it will be very available to memory. Again, the availability heuristic will cause you to misperceive the frequency of these types of events.

Most interestingly, information that is unusual is more memorable. Suppose we give you the following list of words to remember: box, flower, letter, platypus, oven, boat, newspaper, purse, drum, car. Very likely, the easiest word to remember would be platypus, the unusual one. The same thing occurs with memories of events. An event may be available to memory because it is unusual, yet the availability heuristic leads us to judge that the event is common. Did you catch that? In these cases, the availability heuristic makes us think the exact opposite of the true frequency. We end up thinking something is common because it is unusual (and therefore memorable). Yikes.

The misapplication of the availability heuristic sometimes has unfortunate results. For example, if you went to K-12 school in the US over the past 10 years, it is extremely likely that you have participated in lockdown and active shooter drills. Of course, everyone is trying to prevent the tragedy of another school shooting. And believe us, we are not trying to minimize how terrible the tragedy is. But the truth of the matter is, school shootings are extremely rare. Because the federal government does not keep a database of school shootings, the Washington Post has maintained their own running tally. Between 1999 and January 2020 (the date of the most recent school shooting with a death in the US at of the time this paragraph was written), the Post reported a total of 254 people died in school shootings in the US. Not 254 per year, 254 total. That is an average of 12 per year. Of course, that is 254 people who should not have died (particularly because many were children), but in a country with approximately 60,000,000 students and teachers, this is a very small risk.

But many students and teachers are terrified that they will be victims of school shootings because of the availability heuristic. It is so easy to think of examples (they are very available to memory) that people believe the event is very common. It is not. And there is a downside to this. We happen to believe that there is an enormous gun violence problem in the United States. According the the Centers for Disease Control and Prevention, there were 39,773 firearm deaths in the US in 2017. Fifteen of those deaths were in school shootings, according to the Post. 60% of those deaths were suicides. When people pay attention to the school shooting risk (low), they often fail to notice the much larger risk.

And examples like this are by no means unique. The authors of this book have been teaching psychology since the 1990’s. We have been able to make the exact same arguments about the misapplication of the availability heuristics and keep them current by simply swapping out for the “fear of the day.” In the 1990’s it was children being kidnapped by strangers (it was known as “stranger danger”) despite the facts that kidnappings accounted for only 2% of the violent crimes committed against children, and only 24% of kidnappings are committed by strangers (US Department of Justice, 2007). This fear overlapped with the fear of terrorism that gripped the country after the 2001 terrorist attacks on the World Trade Center and US Pentagon and still plagues the population of the US somewhat in 2020. After a well-publicized, sensational act of violence, people are extremely likely to increase their estimates of the chances that they, too, will be victims of terror. Think about the reality, however. In October of 2001, a terrorist mailed anthrax spores to members of the US government and a number of media companies. A total of five people died as a result of this attack. The nation was nearly paralyzed by the fear of dying from the attack; in reality the probability of an individual person dying was 0.00000002.

The availability heuristic can lead you to make incorrect judgments in a school setting as well. For example, suppose you are trying to decide if you should take a class from a particular math professor. You might try to make a judgment of how good a teacher she is by recalling instances of friends and acquaintances making comments about her teaching skill. You may have some examples that suggest that she is a poor teacher very available to memory, so on the basis of the availability heuristic you judge her a poor teacher and decide to take the class from someone else. What if, however, the instances you recalled were all from the same person, and this person happens to be a very colorful storyteller? The subsequent ease of remembering the instances might not indicate that the professor is a poor teacher after all.

Although the availability heuristic is obviously important, it is not the only judgment heuristic we use. Amos Tversky and Daniel Kahneman examined the role of heuristics in inductive reasoning in a long series of studies. Kahneman received a Nobel Prize in Economics for this research in 2002, and Tversky would have certainly received one as well if he had not died of melanoma at age 59 in 1996 (Nobel Prizes are not awarded posthumously). Kahneman and Tversky demonstrated repeatedly that people do not reason in ways that are consistent with the laws of probability. They identified several heuristic strategies that people use instead to make judgments about likelihood. The importance of this work for economics (and the reason that Kahneman was awarded the Nobel Prize) is that earlier economic theories had assumed that people do make judgments rationally, that is, in agreement with the laws of probability.

Another common heuristic that people use for making judgments is the  representativeness heuristic (Kahneman & Tversky 1973). Suppose we describe a person to you. He is quiet and shy, has an unassuming personality, and likes to work with numbers. Is this person more likely to be an accountant or an attorney? If you said accountant, you were probably using the representativeness heuristic. Our imaginary person is judged likely to be an accountant because he resembles, or is representative of the concept of, an accountant. When research participants are asked to make judgments such as these, the only thing that seems to matter is the representativeness of the description. For example, if told that the person described is in a room that contains 70 attorneys and 30 accountants, participants will still assume that he is an accountant.

inductive reasoning :  a type of reasoning in which we make judgments about likelihood from sets of evidence

inductively strong argument :  an inductive argument in which the beginning statements lead to a conclusion that is probably true

heuristic :  a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

availability heuristic :  judging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

representativeness heuristic:   judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

Type 1 thinking : fast, automatic, and emotional thinking.

Type 2 thinking : slow, effortful, and logical thinking.

• What percentage of workplace homicides are co-worker violence?

Many people get these questions wrong. The answers are 10%; stairs; skin; 6%. How close were your answers? Explain how the availability heuristic might have led you to make the incorrect judgments.

• Can you think of some other judgments that you have made (or beliefs that you have) that might have been influenced by the availability heuristic?

7.3 Problem Solving

• Please take a few minutes to list a number of problems that you are facing right now.
• Now write about a problem that you recently solved.
• What is your definition of a problem?

Mary has a problem. Her daughter, ordinarily quite eager to please, appears to delight in being the last person to do anything. Whether getting ready for school, going to piano lessons or karate class, or even going out with her friends, she seems unwilling or unable to get ready on time. Other people have different kinds of problems. For example, many students work at jobs, have numerous family commitments, and are facing a course schedule full of difficult exams, assignments, papers, and speeches. How can they find enough time to devote to their studies and still fulfill their other obligations? Speaking of students and their problems: Show that a ball thrown vertically upward with initial velocity v0 takes twice as much time to return as to reach the highest point (from Spiegel, 1981).

These are three very different situations, but we have called them all problems. What makes them all the same, despite the differences? A psychologist might define a  problem   as a situation with an initial state, a goal state, and a set of possible intermediate states. Somewhat more meaningfully, we might consider a problem a situation in which you are in here one state (e.g., daughter is always late), you want to be there in another state (e.g., daughter is not always late), and with no obvious way to get from here to there. Defined this way, each of the three situations we outlined can now be seen as an example of the same general concept, a problem. At this point, you might begin to wonder what is not a problem, given such a general definition. It seems that nearly every non-routine task we engage in could qualify as a problem. As long as you realize that problems are not necessarily bad (it can be quite fun and satisfying to rise to the challenge and solve a problem), this may be a useful way to think about it.

Can we identify a set of problem-solving skills that would apply to these very different kinds of situations? That task, in a nutshell, is a major goal of this section. Let us try to begin to make sense of the wide variety of ways that problems can be solved with an important observation: the process of solving problems can be divided into two key parts. First, people have to notice, comprehend, and represent the problem properly in their minds (called  problem representation ). Second, they have to apply some kind of solution strategy to the problem. Psychologists have studied both of these key parts of the process in detail.

When you first think about the problem-solving process, you might guess that most of our difficulties would occur because we are failing in the second step, the application of strategies. Although this can be a significant difficulty much of the time, the more important source of difficulty is probably problem representation. In short, we often fail to solve a problem because we are looking at it, or thinking about it, the wrong way.

problem :  a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

problem representation :  noticing, comprehending and forming a mental conception of a problem

Defining and Mentally Representing Problems in Order to Solve Them

So, the main obstacle to solving a problem is that we do not clearly understand exactly what the problem is. Recall the problem with Mary’s daughter always being late. One way to represent, or to think about, this problem is that she is being defiant. She refuses to get ready in time. This type of representation or definition suggests a particular type of solution. Another way to think about the problem, however, is to consider the possibility that she is simply being sidetracked by interesting diversions. This different conception of what the problem is (i.e., different representation) suggests a very different solution strategy. For example, if Mary defines the problem as defiance, she may be tempted to solve the problem using some kind of coercive tactics, that is, to assert her authority as her mother and force her to listen. On the other hand, if Mary defines the problem as distraction, she may try to solve it by simply removing the distracting objects.

As you might guess, when a problem is represented one way, the solution may seem very difficult, or even impossible. Seen another way, the solution might be very easy. For example, consider the following problem (from Nasar, 1998):

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 miles per hour. At the same time, a fly that travels at a steady 15 miles per hour starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner until he is crushed between the two front wheels. Question: what total distance did the fly cover?

Please take a few minutes to try to solve this problem.

Most people represent this problem as a question about a fly because, well, that is how the question is asked. The solution, using this representation, is to figure out how far the fly travels on the first leg of its journey, then add this total to how far it travels on the second leg of its journey (when it turns around and returns to the first bicycle), then continue to add the smaller distance from each leg of the journey until you converge on the correct answer. You would have to be quite skilled at math to solve this problem, and you would probably need some time and pencil and paper to do it.

If you consider a different representation, however, you can solve this problem in your head. Instead of thinking about it as a question about a fly, think about it as a question about the bicycles. They are 20 miles apart, and each is traveling 10 miles per hour. How long will it take for the bicycles to reach each other? Right, one hour. The fly is traveling 15 miles per hour; therefore, it will travel a total of 15 miles back and forth in the hour before the bicycles meet. Represented one way (as a problem about a fly), the problem is quite difficult. Represented another way (as a problem about two bicycles), it is easy. Changing your representation of a problem is sometimes the best—sometimes the only—way to solve it.

Unfortunately, however, changing a problem’s representation is not the easiest thing in the world to do. Often, problem solvers get stuck looking at a problem one way. This is called  fixation . Most people who represent the preceding problem as a problem about a fly probably do not pause to reconsider, and consequently change, their representation. A parent who thinks her daughter is being defiant is unlikely to consider the possibility that her behavior is far less purposeful.

Problem-solving fixation was examined by a group of German psychologists called Gestalt psychologists during the 1930’s and 1940’s. Karl Dunker, for example, discovered an important type of failure to take a different perspective called  functional fixedness . Imagine being a participant in one of his experiments. You are asked to figure out how to mount two candles on a door and are given an assortment of odds and ends, including a small empty cardboard box and some thumbtacks. Perhaps you have already figured out a solution: tack the box to the door so it forms a platform, then put the candles on top of the box. Most people are able to arrive at this solution. Imagine a slight variation of the procedure, however. What if, instead of being empty, the box had matches in it? Most people given this version of the problem do not arrive at the solution given above. Why? Because it seems to people that when the box contains matches, it already has a function; it is a matchbox. People are unlikely to consider a new function for an object that already has a function. This is functional fixedness.

Mental set is a type of fixation in which the problem solver gets stuck using the same solution strategy that has been successful in the past, even though the solution may no longer be useful. It is commonly seen when students do math problems for homework. Often, several problems in a row require the reapplication of the same solution strategy. Then, without warning, the next problem in the set requires a new strategy. Many students attempt to apply the formerly successful strategy on the new problem and therefore cannot come up with a correct answer.

The thing to remember is that you cannot solve a problem unless you correctly identify what it is to begin with (initial state) and what you want the end result to be (goal state). That may mean looking at the problem from a different angle and representing it in a new way. The correct representation does not guarantee a successful solution, but it certainly puts you on the right track.

A bit more optimistically, the Gestalt psychologists discovered what may be considered the opposite of fixation, namely  insight . Sometimes the solution to a problem just seems to pop into your head. Wolfgang Kohler examined insight by posing many different problems to chimpanzees, principally problems pertaining to their acquisition of out-of-reach food. In one version, a banana was placed outside of a chimpanzee’s cage and a short stick inside the cage. The stick was too short to retrieve the banana, but was long enough to retrieve a longer stick also located outside of the cage. This second stick was long enough to retrieve the banana. After trying, and failing, to reach the banana with the shorter stick, the chimpanzee would try a couple of random-seeming attempts, react with some apparent frustration or anger, then suddenly rush to the longer stick, the correct solution fully realized at this point. This sudden appearance of the solution, observed many times with many different problems, was termed insight by Kohler.

Lest you think it pertains to chimpanzees only, Karl Dunker demonstrated that children also solve problems through insight in the 1930s. More importantly, you have probably experienced insight yourself. Think back to a time when you were trying to solve a difficult problem. After struggling for a while, you gave up. Hours later, the solution just popped into your head, perhaps when you were taking a walk, eating dinner, or lying in bed.

fixation :  when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

functional fixedness :  a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

mental set :  a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

insight :  a sudden realization of a solution to a problem

Solving Problems by Trial and Error

Correctly identifying the problem and your goal for a solution is a good start, but recall the psychologist’s definition of a problem: it includes a set of possible intermediate states. Viewed this way, a problem can be solved satisfactorily only if one can find a path through some of these intermediate states to the goal. Imagine a fairly routine problem, finding a new route to school when your ordinary route is blocked (by road construction, for example). At each intersection, you may turn left, turn right, or go straight. A satisfactory solution to the problem (of getting to school) is a sequence of selections at each intersection that allows you to wind up at school.

If you had all the time in the world to get to school, you might try choosing intermediate states randomly. At one corner you turn left, the next you go straight, then you go left again, then right, then right, then straight. Unfortunately, trial and error will not necessarily get you where you want to go, and even if it does, it is not the fastest way to get there. For example, when a friend of ours was in college, he got lost on the way to a concert and attempted to find the venue by choosing streets to turn onto randomly (this was long before the use of GPS). Amazingly enough, the strategy worked, although he did end up missing two out of the three bands who played that night.

Trial and error is not all bad, however. B.F. Skinner, a prominent behaviorist psychologist, suggested that people often behave randomly in order to see what effect the behavior has on the environment and what subsequent effect this environmental change has on them. This seems particularly true for the very young person. Picture a child filling a household’s fish tank with toilet paper, for example. To a child trying to develop a repertoire of creative problem-solving strategies, an odd and random behavior might be just the ticket. Eventually, the exasperated parent hopes, the child will discover that many of these random behaviors do not successfully solve problems; in fact, in many cases they create problems. Thus, one would expect a decrease in this random behavior as a child matures. You should realize, however, that the opposite extreme is equally counterproductive. If the children become too rigid, never trying something unexpected and new, their problem solving skills can become too limited.

Effective problem solving seems to call for a happy medium that strikes a balance between using well-founded old strategies and trying new ground and territory. The individual who recognizes a situation in which an old problem-solving strategy would work best, and who can also recognize a situation in which a new untested strategy is necessary is halfway to success.

Solving Problems with Algorithms and Heuristics

For many problems there is a possible strategy available that will guarantee a correct solution. For example, think about math problems. Math lessons often consist of step-by-step procedures that can be used to solve the problems. If you apply the strategy without error, you are guaranteed to arrive at the correct solution to the problem. This approach is called using an  algorithm , a term that denotes the step-by-step procedure that guarantees a correct solution. Because algorithms are sometimes available and come with a guarantee, you might think that most people use them frequently. Unfortunately, however, they do not. As the experience of many students who have struggled through math classes can attest, algorithms can be extremely difficult to use, even when the problem solver knows which algorithm is supposed to work in solving the problem. In problems outside of math class, we often do not even know if an algorithm is available. It is probably fair to say, then, that algorithms are rarely used when people try to solve problems.

Because algorithms are so difficult to use, people often pass up the opportunity to guarantee a correct solution in favor of a strategy that is much easier to use and yields a reasonable chance of coming up with a correct solution. These strategies are called  problem solving heuristics . Similar to what you saw in section 6.2 with reasoning heuristics, a problem solving heuristic is a shortcut strategy that people use when trying to solve problems. It usually works pretty well, but does not guarantee a correct solution to the problem. For example, one problem solving heuristic might be “always move toward the goal” (so when trying to get to school when your regular route is blocked, you would always turn in the direction you think the school is). A heuristic that people might use when doing math homework is “use the same solution strategy that you just used for the previous problem.”

By the way, we hope these last two paragraphs feel familiar to you. They seem to parallel a distinction that you recently learned. Indeed, algorithms and problem-solving heuristics are another example of the distinction between Type 1 thinking and Type 2 thinking.

Although it is probably not worth describing a large number of specific heuristics, two observations about heuristics are worth mentioning. First, heuristics can be very general or they can be very specific, pertaining to a particular type of problem only. For example, “always move toward the goal” is a general strategy that you can apply to countless problem situations. On the other hand, “when you are lost without a functioning gps, pick the most expensive car you can see and follow it” is specific to the problem of being lost. Second, all heuristics are not equally useful. One heuristic that many students know is “when in doubt, choose c for a question on a multiple-choice exam.” This is a dreadful strategy because many instructors intentionally randomize the order of answer choices. Another test-taking heuristic, somewhat more useful, is “look for the answer to one question somewhere else on the exam.”

You really should pay attention to the application of heuristics to test taking. Imagine that while reviewing your answers for a multiple-choice exam before turning it in, you come across a question for which you originally thought the answer was c. Upon reflection, you now think that the answer might be b. Should you change the answer to b, or should you stick with your first impression? Most people will apply the heuristic strategy to “stick with your first impression.” What they do not realize, of course, is that this is a very poor strategy (Lilienfeld et al, 2009). Most of the errors on exams come on questions that were answered wrong originally and were not changed (so they remain wrong). There are many fewer errors where we change a correct answer to an incorrect answer. And, of course, sometimes we change an incorrect answer to a correct answer. In fact, research has shown that it is more common to change a wrong answer to a right answer than vice versa (Bruno, 2001).

The belief in this poor test-taking strategy (stick with your first impression) is based on the  confirmation bias   (Nickerson, 1998; Wason, 1960). You first saw the confirmation bias in Module 1, but because it is so important, we will repeat the information here. People have a bias, or tendency, to notice information that confirms what they already believe. Somebody at one time told you to stick with your first impression, so when you look at the results of an exam you have taken, you will tend to notice the cases that are consistent with that belief. That is, you will notice the cases in which you originally had an answer correct and changed it to the wrong answer. You tend not to notice the other two important (and more common) cases, changing an answer from wrong to right, and leaving a wrong answer unchanged.

Because heuristics by definition do not guarantee a correct solution to a problem, mistakes are bound to occur when we employ them. A poor choice of a specific heuristic will lead to an even higher likelihood of making an error.

algorithm :  a step-by-step procedure that guarantees a correct solution to a problem

problem solving heuristic :  a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

confirmation bias :  people’s tendency to notice information that confirms what they already believe

An Effective Problem-Solving Sequence

You may be left with a big question: If algorithms are hard to use and heuristics often don’t work, how am I supposed to solve problems? Robert Sternberg (1996), as part of his theory of what makes people successfully intelligent (Module 8) described a problem-solving sequence that has been shown to work rather well:

• Identify the existence of a problem.  In school, problem identification is often easy; problems that you encounter in math classes, for example, are conveniently labeled as problems for you. Outside of school, however, realizing that you have a problem is a key difficulty that you must get past in order to begin solving it. You must be very sensitive to the symptoms that indicate a problem.
• Define the problem.  Suppose you realize that you have been having many headaches recently. Very likely, you would identify this as a problem. If you define the problem as “headaches,” the solution would probably be to take aspirin or ibuprofen or some other anti-inflammatory medication. If the headaches keep returning, however, you have not really solved the problem—likely because you have mistaken a symptom for the problem itself. Instead, you must find the root cause of the headaches. Stress might be the real problem. For you to successfully solve many problems it may be necessary for you to overcome your fixations and represent the problems differently. One specific strategy that you might find useful is to try to define the problem from someone else’s perspective. How would your parents, spouse, significant other, doctor, etc. define the problem? Somewhere in these different perspectives may lurk the key definition that will allow you to find an easier and permanent solution.
• Formulate strategy.  Now it is time to begin planning exactly how the problem will be solved. Is there an algorithm or heuristic available for you to use? Remember, heuristics by their very nature guarantee that occasionally you will not be able to solve the problem. One point to keep in mind is that you should look for long-range solutions, which are more likely to address the root cause of a problem than short-range solutions.
• Represent and organize information.  Similar to the way that the problem itself can be defined, or represented in multiple ways, information within the problem is open to different interpretations. Suppose you are studying for a big exam. You have chapters from a textbook and from a supplemental reader, along with lecture notes that all need to be studied. How should you (represent and) organize these materials? Should you separate them by type of material (text versus reader versus lecture notes), or should you separate them by topic? To solve problems effectively, you must learn to find the most useful representation and organization of information.
• Allocate resources.  This is perhaps the simplest principle of the problem solving sequence, but it is extremely difficult for many people. First, you must decide whether time, money, skills, effort, goodwill, or some other resource would help to solve the problem Then, you must make the hard choice of deciding which resources to use, realizing that you cannot devote maximum resources to every problem. Very often, the solution to problem is simply to change how resources are allocated (for example, spending more time studying in order to improve grades).
• Monitor and evaluate solutions.  Pay attention to the solution strategy while you are applying it. If it is not working, you may be able to select another strategy. Another fact you should realize about problem solving is that it never does end. Solving one problem frequently brings up new ones. Good monitoring and evaluation of your problem solutions can help you to anticipate and get a jump on solving the inevitable new problems that will arise.

Please note that this as  an  effective problem-solving sequence, not  the  effective problem solving sequence. Just as you can become fixated and end up representing the problem incorrectly or trying an inefficient solution, you can become stuck applying the problem-solving sequence in an inflexible way. Clearly there are problem situations that can be solved without using these skills in this order.

Additionally, many real-world problems may require that you go back and redefine a problem several times as the situation changes (Sternberg et al. 2000). For example, consider the problem with Mary’s daughter one last time. At first, Mary did represent the problem as one of defiance. When her early strategy of pleading and threatening punishment was unsuccessful, Mary began to observe her daughter more carefully. She noticed that, indeed, her daughter’s attention would be drawn by an irresistible distraction or book. Fresh with a re-representation of the problem, she began a new solution strategy. She began to remind her daughter every few minutes to stay on task and remind her that if she is ready before it is time to leave, she may return to the book or other distracting object at that time. Fortunately, this strategy was successful, so Mary did not have to go back and redefine the problem again.

Pick one or two of the problems that you listed when you first started studying this section and try to work out the steps of Sternberg’s problem solving sequence for each one.

a mental representation of a category of things in the world

an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

knowledge about one’s own cognitive processes; thinking about your thinking

individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

Thinking like a scientist in your everyday life for the purpose of drawing correct conclusions. It entails skepticism; an ability to identify biases, distortions, omissions, and assumptions; and excellent deductive and inductive reasoning, and problem solving skills.

a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

an inclination, tendency, leaning, or prejudice

a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

a set of statements in which the beginning statements lead to a conclusion

an argument for which true beginning statements guarantee that the conclusion is true

a type of reasoning in which we make judgments about likelihood from sets of evidence

an inductive argument in which the beginning statements lead to a conclusion that is probably true

fast, automatic, and emotional thinking

slow, effortful, and logical thinking

a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

udging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

noticing, comprehending and forming a mental conception of a problem

when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

a sudden realization of a solution to a problem

a step-by-step procedure that guarantees a correct solution to a problem

The tendency to notice and pay attention to information that confirms your prior beliefs and to ignore information that disconfirms them.

a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

Introduction to Psychology Copyright © 2020 by Ken Gray; Elizabeth Arnott-Hill; and Or'Shaundra Benson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Curtis Silver

The Importance of Logic and Critical Thinking

"Critical thinking is a desire to seek, patience to doubt, fondness to meditate, slowness to assert, readiness to consider, carefulness to dispose and set in order; and hatred for every kind of imposture." - Francis Bacon (1605)

As parents, we are tasked with instilling a plethora of different values into our children. While some parents in the world choose to instill a lack of values in their kids, those of us that don't want our children growing up to be criminals and various misfits try a bit harder. Values and morality are one piece of the pie. These are important things to mold into a child's mind, but there are also other items in life to focus on as well. It starts with looking both ways to cross the street and either progresses from there, or stops.

If you stopped explaining the world to your children after they learned to cross the street, then perhaps you should stop reading and go back to surfing for funny pictures of cats. I may use some larger words that you might not understand, making you angry and causing you to leave troll-like comments full of bad grammar and moronic thought processes. However, if you looked at the crossing the street issue as I did – as a logical problem with cause and effect and a probable solution – then carry on. You are my target audience.

Or perhaps the opposite is true, as the former are the people that could benefit from letting some critical thinking into their lives. So what exactly is critical thinking? This bit by Linda Elder in a paper on CriticalThinking.org pretty much sums it up:

Through critical thinking, as I understand it, we acquire a means of assessing and upgrading our ability to judge well. It enables us to go into virtually any situation and to figure out the logic of whatever is happening in that situation. It provides a way for us to learn from new experiences through the process of continual self-assessment. Critical thinking, then, enables us to form sound beliefs and judgments, and in doing so, provides us with a basis for a 'rational and reasonable' emotional life. — Inquiry: Critical Thinking Across the Disciplines, Winter, 1996. Vol. XVI, No. 2.

The rationality of the world is what is at risk. Too many people are taken advantage of because of their lack of critical thinking, logic and deductive reasoning. These same people are raising children without these same skills, creating a whole new generation of clueless people.

By Matt Burgess

By Andy Greenberg

By Matthew Gault

To wit, a personal tale of deductive reasoning:

Recently I needed a new transmission for the family van. The warranty on the power train covers the transmission up to 100,000 miles. The van has around 68,000 miles on it. Therefore, even the logic-less dimwit could easily figure that the transmission was covered. Well, this was true until the dealership told me that it wasn't, stating that because we didn't get the scheduled transmission service (which is basically a fluid change) at 30,000 and 60,000 miles the warranty was no longer valid. Now, there are many people that would argue this point, but many more that would shrug, panic, and accept the full cost of repairs.

I read the warranty book. I had a receipt that said the fluid was checked at 60,000 but not replaced. A friend on Twitter pointed out the fact that they were using 100,000 mile transmission fluid. So logically, the fluid would not have to be replaced under 100,000 miles if it wasn't needed, right? So why the stipulation that it needed to be replaced at 60,000 and the loose assumption that not doing that would void the warranty? So I asked the warranty guy to show me in the book where the two items are related. Where it explicitly says that if you don't get the service, the transmission isn't covered. There were portions where it said the service was recommended, but never connecting to actual repairs. Finally the warranty guy shrugged, admitted I was right and said the service was covered.

In this case, valid logic equaled truth and a sound argument. I used very simple reasoning and logic to determine that I was being inadvertently screwed. I say "inadvertently" because I truly believe based on their behavior that they were not intentionally trying to screw me. They believed the two items were related, they had had this argument many times before and were not prepared to be questioned. While both the service manager and the warranty guy seemed at least junior college educated, proving my argument to them took longer than it should have between three adults.

However, valid logic does not always guarantee truth or a sound argument. This is where it gets a little funky. Valid logic is when the structure of logic is correct in the way of syntax and semantics rather than truth. Truth comes from deductive reasoning of said logic. For example:

All transmissions are covered parts. All covered parts are free. Therefore, all transmissions are free. This logic is technically valid, and if the premises are true, then of course the conclusion must be true. You can see here however that it's not always true, though in some situations it could be. While the logic is valid, not all transmissions are free, only those covered by the warranty. So based on that, saying all transmissions are free is not sound logic.

To take it one step further:

All Daleks are brown. Some brown things are Cylons . Therefore, some Daleks are Cylons. Sci-fi fan or not, you probably know that this is not true. The basic lesson here is that, while the logic above might seem valid because of the structure of the statement, it takes a further understanding to figure out why it's not necessarily true: That is, based on the first two statements it's possible that some Daleks are Cylons, but it's not logically concludable. That's where deductive reasoning comes on top of the logic. The underlying lesson here is not to immediately assume everything you read or are told is true, something all children need to and should learn.

This is the direct lesson that needs to be passed on to our children: that of not accepting the immediately visible logic. While not all problems are complex enough to require the scientific method, some of them need some deduction to determine if they are true. Take the example above — how many kids would immediately be satisfied with the false conclusion? Sure, it's a bit geeky with the examples, but switch out bears for Daleks and puppies for Cylons. That makes it easier, and takes the actual research out of it (to find out what Daleks and Cylons are respectively) but many people would just accept that in fact some bears are puppies, if presented with this problem in the context of a textbook or word problem.

Maybe I'm being paranoid or thinking too doomsday, whatever, but I think this is an epidemic. Children are becoming lazier and not as self sufficient because their parents have a problem with watching a three year old cry after they tell her to remove her own jeans, or ask her to put away her own toys (yes, organizational logic falls under the main topic). These are the same parents who do their kid's science project while the kid is playing video games. These kids grow up lacking the simple problem solving skills that make navigating life much easier. Remember when you were growing up and you had the plastic stacking toys ? Well, instead of toys for early development like that, parents are just plopping their kids down in front of the television. While there is some educational type programming on television, it's just not the same as hands-on experience.

My father is an engineer, and he taught me logic and reasoning by making me solve simple, then complex, problems on my own. Or at least giving me the opportunity to solve them on my own. This helped develop critical thinking and problem solving skills, something a lot of children lack these days. Too often I see children that are not allowed to solve problems on their own; instead their parents simply do it for them without argument or discussion. Hell, I am surrounded by adults every day that are unable to solve simple problems, instead choosing to immediately ask me at which point I have to fill the role that their parents never did and – knowing the solution – tell them to solve it themselves, or at least try first.

One of the things I like to work on with my kids is math. There is nothing that teaches deductive reasoning and logic better than math word problems. They are at the age where basic algebra can come into play, which sharpens their reasoning skills because they start to view real world issues with algebraic solutions. Another thing is logic puzzles , crossword puzzles and first person shooters. Actually, not that last one. That's just the reward.

Since I weeded out the folks that don't teach their kids logic in the first two paragraphs, as representatives of the real world it's up to the rest of us to spread the knowledge. It won't be easy. The best thing we can do is teach these thought processes to our children, so that they may look at other children with looks of bewilderment when other children are unable to solve simple tasks. Hopefully, they will not simply do the task for them, but teach them to think. I'm not saying we need to build a whole new generation of project managers and analysts, but it would be better than a generation of task-oriented mindless office drones with untied shoelaces, shoving on a door at the Midvale School for the Gifted .

h/t to @aubreygirl22 for the logical conversation. Image: Flickr user William Notowidagdo. Used under Creative Commons License.

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25 Logic Puzzles That Will Totally Blow Your Mind, But Also Prove You’re Kind of a Genius

• Author: Parade

Logic Puzzles

Logic puzzles may fall under the category of math , but they are true works of art. These word problems test your mind power and inspire you to think harder than you’ve ever thought before. Once you start solving these brain teasers , though, you’ll start to see common patterns and themes: how to cross rivers, cheat death, and tell who is lying.

Although these logic puzzles for adults can be solved by complicated mathematical equations, they can also be thought through in your head. Don’t worry, we’ll start you off with easy logic puzzles and always provide explanations for the answer; but be warned: Even after you get good at them, some of these hard logic puzzles and problems could have you stumped for hours. Ready to take the challenge?

Easy Logic Puzzles

1. Logic Puzzle: There are two ducks in front of a duck, two ducks behind a duck and a duck in the middle. How many ducks are there?

Answer: Three. Two ducks are in front of the last duck; the first duck has two ducks behind; one duck is between the other two.

2. Logic Puzzle: Five people were eating apples , A finished before B, but behind C. D finished before E, but behind B. What was the finishing order?

Answer: CABDE. Putting the first three in order, A finished in front of B but behind C, so CAB. Then, we know D finished before B, so CABD. We know E finished after D, so CABDE.

3. Logic Puzzle: Jack is looking at Anne. Anne is looking at George. Jack is married, George is not, and we don’t know if Anne is married. Is a married person looking at an unmarried person?

Answer: Yes. If Anne is married, then she is married and looking at George, who is unmarried. If Anne is unmarried, then Jack, who is married, is looking at her. Either way, the statement is correct.

4. Logic Puzzle: A man has 53 socks in his drawer: 21 identical blue, 15 identical black and 17 identical red. The lights are out and he is completely in the dark. How many socks must he take out to make 100 percent certain he has at least one pair of black socks?

Answer: 40 socks. If he takes out 38 socks (adding the two biggest amounts, 21 and 17), although it is very unlikely, it is possible they could all be blue and red. To make 100 percent certain that he also has a pair of black socks he must take out a further two socks.

5. Logic Puzzle: The day before two days after the day before tomorrow is Saturday. What day is it today?

Answer: Friday . The “day before tomorrow” is today; “the day before two days after” is really one day after. So if “one day after today is Saturday,” then it must be Friday.

6. Logic Puzzle: This “burning rope” problem is a classic logic puzzle. You have two ropes that each take an hour to burn, but burn at inconsistent rates. How can you measure 45 minutes? (You can light one or both ropes at one or both ends at the same time.)

Answer: Because they both burn inconsistently, you can’t just light one end of a rope and wait until it’s 75 percent of the way through. But, this is what you can do: Light the first rope at both ends, and light the other rope at one end, all at the same time. The first rope will take 30 minutes to burn (even if one side burns faster than the other, it still takes 30 minutes). The moment the first rope goes out, light the other end of the second rope. Because the time elapsed of the second rope burning was 30 minutes, the remaining rope will also take 30 minutes; lighting it from both ends will cut that in half to 15 minutes, giving you 45 minutes all together.

Related: Trivia Questions for Kids

Lying or telling the truth logic puzzles

7. Logic Puzzle: You’re at a fork in the road in which one direction leads to the City of Lies (where everyone always lies) and the other to the City of Truth (where everyone always tells the truth). There’s a person at the fork who lives in one of the cities, but you’re not sure which one. What question could you ask the person to find out which road leads to the City of Truth?

Answer: “Which direction do you live?” Someone from the City of Lies will lie and point to the City of Truth; someone from the City of Truth would tell the truth and also point to the City of Truth.

8. Logic Puzzle: A girl meets a lion and unicorn in the forest. The lion lies every Monday, Tuesday and Wednesday and the other days he speaks the truth. The unicorn lies on Thursdays, Fridays and Saturdays, and the other days of the week he speaks the truth. “Yesterday I was lying,” the lion told the girl. “So was I,” said the unicorn. What day is it?

Answer:  Thursday. The only day they both tell the truth is Sunday; but today can’t be Sunday because the lion also tells the truth on Saturday (yesterday). Going day by day, the only day one of them is lying and one of them is telling the truth with those two statements is Thursday.

9. Logic Puzzle: There are three people (Alex, Ben and Cody), one of whom is a knight, one a knave, and one a spy. The knight always tells the truth, the knave always lies, and the spy can either lie or tell the truth. Alex says: "Cody is a knave.” Ben says: "Alex is a knight.” Cody says: "I am the spy.” Who is the knight, who the knave, and who the spy?

Answer: We know Ben isn’t telling the truth because if he was, there would be two knights; so Ben could be either the knave or the spy. Cody also can’t be the knight, because then his statement would be a lie. So that must mean Alex is the knight. Ben, therefore, must be the spy, since the spy sometimes tells the truth; leaving Cody as the knave.

River crossing logic puzzles

10. Logic Puzzle: A farmer wants to cross a river and take with him a wolf , a goat and a cabbage . He has a boat, but it can only fit himself plus either the wolf, the goat or the cabbage. If the wolf and the goat are alone on one shore, the wolf will eat the goat. If the goat and the cabbage are alone on the shore, the goat will eat the cabbage. How can the farmer bring the wolf, the goat and the cabbage across the river without anything being eaten?

Answer: First, the farmer takes the goat across. The farmer returns alone and then takes the wolf across, but returns with the goat. Then the farmer takes the cabbage across, leaving it with the wolf and returning alone to get the goat.

11. Logic Puzzle: Let’s pretend we’re on the metric system and use kilograms instead of pounds to give us a starting base number of 100. Four people (Alex, Brook, Chris and Dusty) want to cross a river in a boat that can only carry 100kg. Alex weighs 90kg, Brook weighs 80kg, Chris weighs 60kg and Dusty weighs 40kg, and they have 20kg of supplies. How do they get across?

Answer: There may be a couple variations that will work, but here’s one way: Chris and Dusty row across (combined 100kg), Dusty returns. Alex rows over, and Chris returns. Chris and Dusty row across again, Dusty returns. Brook rows across with the supplies (combined 100kg), and Chris returns. Chris and Dusty row across again.

12. Logic Puzzle: This famous river crossing problem is known as the “bridge and torch” puzzle. Four people are crossing a bridge at night, so they all need a torch—but they just have one that only lasts 15 minutes. Alice can cross in one minute, Ben in two minutes, Cindy in five minutes and Don in eight minutes. No more than two people can cross at a time; and when two cross, they have to go at the slower person’s pace. How do they get across in 15 minutes?

Answer: Alice and Ben cross first in two minutes, and Alice crosses back alone with the torch in one minute. Then the two slowest people, Cindy and Don, cross in eight minutes. Ben returns in two minutes, and Alice and Ben return in two minutes. They just made it in 15 minutes exactly.

Related: 101 Fun Facts

Deadly choices logic puzzles

13. Logic Puzzle: A bad guy is playing Russian roulette with a six-shooter revolver. He puts in one bullet, spins the chambers and fires at you, but no bullet comes out. He gives you the choice of whether or not he should spin the chambers again before firing a second time. Should he spin again?

Answer: Yes. Before he spins, there’s a one in six chance of a bullet being fired. After he spins, one of those chances has been taken away, leaving a one in five chance and making it more likely a bullet will be fired. Best to spin again.

14. Logic Puzzle: Same situation, but two bullets are put in consecutive chambers. Should you tell the bad guy to spin the chambers again?

Answer: No. With two bullets, you have two chances in six (or one in three) to get hit with a bullet before he fires the first time. Because we know the previous round was one of four empty chambers, that leaves four positions the gun could now be in, with only one followed by a bullet; therefore leaving you with a one in four chance the second round will fire. Since one in four is better odds than one in three, he shouldn’t spin again.

15. Logic Puzzle: This one could also fall in the lying/truth category. A man is caught on the king's property. He is brought before the king to be punished. The king says, "You must give me a statement. If it is true, you will be killed by lions. If it is false, you will be killed by trampling of wild buffalo. If I can’t figure it out, I’ll have to let you go.” Sure enough, the man was released. What was the man's statement?

Answer: "I will be killed by trampling of wild buffalo.” This stumped the king because if it’s true, he’ll be killed by lions, which would render the statement not true. If it’s a lie, he’d be killed by wild buffalo, which would make it a truth. Since the king had no solution, he had to let the man go.

Hard Logic Puzzles for Adults

16. Logic Puzzle: Susan and Lisa decided to play tennis against each other. They bet $1 on each game they played. Susan won three bets and Lisa won $5. How many games did they play?

Answer: Eleven. Because Lisa lost three games to Susan, she had lost $3 ($1 per game). So, she had to win back that $3 with three more games, then win another five games to win $5.

17. Logic Puzzle: If five cats can catch five mice in five minutes, how long will it take one cat to catch one mouse?

Answer: Five minutes. Using the information we know, it would take one cat 25 minutes to catch all five mice (5x5=25). Then working backward and dividing 25 by five, we get five minutes for one cat to catch each mouse.

18. Logic Puzzle: There is a barrel with no lid and some wine in it. “This barrel of wine is more than half full,” says the woman. “No, it's not,” says the man. “It’s less than half full.” Without any measuring implements and without removing any wine from the barrel, how can they easily determine who is correct?

Answer: Tilt the barrel until the wine barely touches the lip of the barrel. If the bottom of the barrel is visible then it is less than half full. If the barrel bottom is still completely covered by the wine, then it is more than half full.

19. Logic Puzzle: There are three bags, each containing two marbles. Bag A contains two white marbles, Bag B contains two black marbles, and Bag C contains one white marble and one black marble. You pick a random bag and take out one marble, which is white. What is the probability that the remaining marble from the same bag is also white?

Answer: 2 out of 3. You know you don’t have Bag B. But because Bag A has two white marbles, you could have picked either marble; if you think of it as four marbles in total from Bags A and C, three white and one black, you’ll have a greater chance of picking another white marble.

20. Logic Puzzle: Three men are lined up behind each other. The tallest man is in the back and can see the heads of the two in front of him; the middle man can see the one man in front of him; the man in front can’t see anyone. They are blindfolded and hats are placed on their heads, picked from three black hats and two white hats. The extra two hats are hidden and the blindfolds removed. The tallest man is asked if he knows what color hat he’s wearing; he doesn’t. The middle man is asked if he knows; he doesn’t. But the man in front, who can’t see anyone, says he knows. How does he know, and what color hat is he wearing?

Answer: Black. The man in front knew he and the middle man aren’t both wearing white hats or the man in the back would have known he had a black hat (since there are only two white hats). The man in front also knows the middle man didn’t see him with a white hat because if he did, based on the tallest man’s answer, the middle man would have known he himself was wearing a black hat. So, the man in front knows his hat must be black.

21. Logic Puzzle: There are three crates, one with apples, one with oranges, and one with both apples and oranges mixed. Each crate is closed and labeled with one of three labels: Apples, Oranges, or Apples and Oranges. The label maker broke and labeled all of the crates incorrectly. How could you pick just one fruit from one crate to figure out what’s in each crate?

Answer: Pick a fruit from the crate marked Apples and Oranges. If that fruit is an apple, you know that the crate should be labeled Apples because all of the labels are incorrect as they are. Therefore, you know the crate marked Apples must be Oranges (if it were labeled Apples and Oranges, the Oranges crate would be labeled correctly, and we know it isn’t), and the one marked Oranges is Apples and Oranges. Alternately, if you picked an orange from the crate marked Apples and Oranges, you know that crate should be marked Oranges, the one marked Oranges must be Apples, and the one marked Apples must be Apples and Oranges.

Hardest Logic Puzzles for Adults

22. Logic Puzzle: A teacher writes six words on a board: “cat dog has max dim tag.” She gives three students, Albert, Bernard and Cheryl each a piece of paper with one letter from one of the words. Then she asks, “Albert, do you know the word?” Albert immediately replies yes. She asks, “Bernard, do you know the word?” He thinks for a moment and replies yes. Then she asks Cheryl the same question. She thinks and then replies yes. What is the word?

Answer: Dog. Albert knows right away because he has one of the unique letters that only appear once in all the words: c o h s x i. So, we know the word is not “tag.” All of these unique letters appear in different words, except for “h” and “s” in “has,” and Bernard can figure out what the word is from the unique letters that are left: t, g, h, s. This eliminates “max” and “dim.” Cheryl can then narrow it down the same way. Because there is only one unique letter left, the letter “d,” the word must be “dog.” (For more on this answer, watch the video below.)

23. Logic Puzzle: You have five boxes in a row numbered 1 to 5, in which a cat is hiding. Every night he jumps to an adjacent box, and every morning you have one chance to open a box to find him. How do you win this game of hide and seek?

Answer: Check boxes 2, 3, and 4 in order until you find him. Here’s why: He’s either in an odd or even-numbered box. If he’s in an even box (box 2 or 4) and you check box 2 and here’s there, great; if not you know he was in box 4, which means the next night he will move to box 3 or 5. The next morning, check box 3; if he’s not there that means he was in box 5 and so the next night he’ll be in box 4, and you’ve got him. If he was in an odd-numbered box to begin with (1, 3, or 5), though, you might not find him in that first round of checking boxes 2, 3 and 4. But if this is the case, you know that on the fourth night he’ll have to be in an even-numbered box (because he switches every night: odd, even, odd, even), so then you can start the process again as described above. This means if you check boxes 2, 3, and 4 in that order, you will find him within two rounds (one round of 2, 3, 4; followed by another round of  2, 3, 4). For more on this answer, watch the video below.

24. Logic Puzzle: The “Monty Hall” problem was made famous when it appeared in Parade magazine’s “ Ask Marilyn ” column in 1990, and it was so counterintuitive it had everyone from high school students to top mathematical minds questioning the answer—but rest assured, the solution is accurate. Named for the Let’s Make a Deal game show host, the puzzle goes like this: You are given three doors to choose from, one of which contains a car and the other two contain goats. After you’ve chosen one but haven’t opened it, Monty, who knows where everything is, reveals the location of a goat from behind one of the other two doors. Should you stick with your original choice or switch, if you want the car?

Answer: You should switch. At the beginning, your choice starts out as a one in three chance of picking the car; the two doors with goats contain 2/3 of the chance. But since Monty knows and shows you where one of the goats is, that 2/3 chance now rests solely with the third door (your choice retains its original 1/3 chance; you were more likely to pick a goat to begin with). So, the odds are better if you switch.

Near Impossible Logic Puzzle for Adults

25. Logic Puzzle: This conundrum, a variation on a lying/truth problem, has famously been called the hardest logic puzzle ever. You meet three gods on a mountain top. One always tells the truth, one always lies, and one tells the truth or lies randomly. We can call them Truth, False and Random. They understand English but answer in their own language, with ja or da for yes and no—but you don’t know which is which. You can ask three questions to any of the gods (and you can ask the same god more than one question), and they will answer with ja or da. What three questions do you ask to figure out who’s who?

Answer: Before getting to the answer, let’s think of a hypothetical question you know the answer to, such as “Does two plus two equal four?” Then, phrase it so you’re asking it as an embedded question: “If I asked you if two plus two equals four, would you answer ja?” If ja means yes, Truth would answer ja, but so would False (he always lies, so he’d say ja even though he really would answer da). If ja means no, they both would still answer ja—in this case, False would answer the embedded question with ja, but saying da to the overall question would be telling the truth, so he says ja. (Random’s answer would be meaningless because we don’t know whether he lies or tells the truth.)

But what if you said, “If I asked you if two plus two equals five, would you answer ja?” If ja means yes, Truth would answer da, as would False; if ja means no, they’d also both answer da. So, you know that if the embedded question is correct, Truth and False always answer with the same word you use; if the embedded question is incorrect, they always answer with the opposite word. You also know they always answer with the same word as each other.

With this reasoning, ask the god in the middle your first question: “If I asked you whether the god on my left is Random, would you answer ja?” If the god answers ja and you’re talking to either Truth or False, following the above logic you know the embedded question is correct, and the god to the left is Random. It’s also possible that you’re speaking to Random; but you know no matter who you’re talking to, the god on the right is not Random. If the answer is da, the opposite is the case, and you know the god on the left isn’t Random. Next, you can ask the god you definitely know isn’t Random a question using the same structure: “If I were to ask you if you are Truth, would you say ja?” If they answer ja, you know you’re talking to Truth; if they answer da you know you’re talking with False. Then once you’ve identified that god as True or False, you can ask the same god a final question to identify Random: “If I asked you if the god in the middle is Random, would you say ja?” By process of elimination, you can then identify the last god.

If you made it this far, you’re a true logic puzzle genius!

Want more fun? Try these 101 Riddles (with Answers) or Best Online Games .

Story by Tina Donvito .

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• The Art of Effective Problem Solving: A Step-by-Step Guide

Daniel Croft

Daniel Croft is an experienced continuous improvement manager with a Lean Six Sigma Black Belt and a Bachelor's degree in Business Management. With more than ten years of experience applying his skills across various industries, Daniel specializes in optimizing processes and improving efficiency. His approach combines practical experience with a deep understanding of business fundamentals to drive meaningful change.

• Last Updated: February 6, 2023
• Learn Lean Sigma
• Problem Solving

Whether we realise it or not, problem solving skills are an important part of our daily lives. From resolving a minor annoyance at home to tackling complex business challenges at work, our ability to solve problems has a significant impact on our success and happiness. However, not everyone is naturally gifted at problem-solving, and even those who are can always improve their skills. In this blog post, we will go over the art of effective problem-solving step by step.

You will learn how to define a problem, gather information, assess alternatives, and implement a solution, all while honing your critical thinking and creative problem-solving skills. Whether you’re a seasoned problem solver or just getting started, this guide will arm you with the knowledge and tools you need to face any challenge with confidence. So let’s get started!

Problem Solving Methodologies

Individuals and organisations can use a variety of problem-solving methodologies to address complex challenges. 8D and A3 problem solving techniques are two popular methodologies in the Lean Six Sigma framework.

Methodology of 8D (Eight Discipline) Problem Solving:

The 8D problem solving methodology is a systematic, team-based approach to problem solving. It is a method that guides a team through eight distinct steps to solve a problem in a systematic and comprehensive manner.

The 8D process consists of the following steps:

• Form a team: Assemble a group of people who have the necessary expertise to work on the problem.
• Define the issue: Clearly identify and define the problem, including the root cause and the customer impact.
• Create a temporary containment plan: Put in place a plan to lessen the impact of the problem until a permanent solution can be found.
• Identify the root cause: To identify the underlying causes of the problem, use root cause analysis techniques such as Fishbone diagrams and Pareto charts.
• Create and test long-term corrective actions: Create and test a long-term solution to eliminate the root cause of the problem.
• Implement and validate the permanent solution: Implement and validate the permanent solution’s effectiveness.
• Prevent recurrence: Put in place measures to keep the problem from recurring.
• Recognize and reward the team: Recognize and reward the team for its efforts.

Download the 8D Problem Solving Template

A3 Problem Solving Method:

The A3 problem solving technique is a visual, team-based problem-solving approach that is frequently used in Lean Six Sigma projects. The A3 report is a one-page document that clearly and concisely outlines the problem, root cause analysis, and proposed solution.

The A3 problem-solving procedure consists of the following steps:

• Determine the issue: Define the issue clearly, including its impact on the customer.
• Perform root cause analysis: Identify the underlying causes of the problem using root cause analysis techniques.
• Create and implement a solution: Create and implement a solution that addresses the problem’s root cause.
• Monitor and improve the solution: Keep an eye on the solution’s effectiveness and make any necessary changes.

Subsequently, in the Lean Six Sigma framework, the 8D and A3 problem solving methodologies are two popular approaches to problem solving. Both methodologies provide a structured, team-based problem-solving approach that guides individuals through a comprehensive and systematic process of identifying, analysing, and resolving problems in an effective and efficient manner.

Step 1 – Define the Problem

The definition of the problem is the first step in effective problem solving. This may appear to be a simple task, but it is actually quite difficult. This is because problems are frequently complex and multi-layered, making it easy to confuse symptoms with the underlying cause. To avoid this pitfall, it is critical to thoroughly understand the problem.

To begin, ask yourself some clarifying questions:

• What exactly is the issue?
• What are the problem’s symptoms or consequences?
• Who or what is impacted by the issue?
• When and where does the issue arise?

Answering these questions will assist you in determining the scope of the problem. However, simply describing the problem is not always sufficient; you must also identify the root cause. The root cause is the underlying cause of the problem and is usually the key to resolving it permanently.

Try asking “why” questions to find the root cause:

• What causes the problem?
• Why does it continue?
• Why does it have the effects that it does?

By repeatedly asking “ why ,” you’ll eventually get to the bottom of the problem. This is an important step in the problem-solving process because it ensures that you’re dealing with the root cause rather than just the symptoms.

Once you have a firm grasp on the issue, it is time to divide it into smaller, more manageable chunks. This makes tackling the problem easier and reduces the risk of becoming overwhelmed. For example, if you’re attempting to solve a complex business problem, you might divide it into smaller components like market research, product development, and sales strategies.

To summarise step 1, defining the problem is an important first step in effective problem-solving. You will be able to identify the root cause and break it down into manageable parts if you take the time to thoroughly understand the problem. This will prepare you for the next step in the problem-solving process, which is gathering information and brainstorming ideas.

Step 2 – Gather Information and Brainstorm Ideas

Gathering information and brainstorming ideas is the next step in effective problem solving. This entails researching the problem and relevant information, collaborating with others, and coming up with a variety of potential solutions. This increases your chances of finding the best solution to the problem.

Begin by researching the problem and relevant information. This could include reading articles, conducting surveys, or consulting with experts. The goal is to collect as much information as possible in order to better understand the problem and possible solutions.

Next, work with others to gather a variety of perspectives. Brainstorming with others can be an excellent way to come up with new and creative ideas. Encourage everyone to share their thoughts and ideas when working in a group, and make an effort to actively listen to what others have to say. Be open to new and unconventional ideas and resist the urge to dismiss them too quickly.

Finally, use brainstorming to generate a wide range of potential solutions. This is the place where you can let your imagination run wild. At this stage, don’t worry about the feasibility or practicality of the solutions; instead, focus on generating as many ideas as possible. Write down everything that comes to mind, no matter how ridiculous or unusual it may appear. This can be done individually or in groups.

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the next step in the problem-solving process, which we’ll go over in greater detail in the following section.

Step 3 – Evaluate Options and Choose the Best Solution

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the third step in effective problem solving, and it entails weighing the advantages and disadvantages of each solution, considering their feasibility and practicability, and selecting the solution that is most likely to solve the problem effectively.

To begin, weigh the advantages and disadvantages of each solution. This will assist you in determining the potential outcomes of each solution and deciding which is the best option. For example, a quick and easy solution may not be the most effective in the long run, whereas a more complex and time-consuming solution may be more effective in solving the problem in the long run.

Consider each solution’s feasibility and practicability. Consider the following:

• Can the solution be implemented within the available resources, time, and budget?
• What are the possible barriers to implementing the solution?
• Is the solution feasible in today’s political, economic, and social environment?

You’ll be able to tell which solutions are likely to succeed and which aren’t by assessing their feasibility and practicability.

Finally, choose the solution that is most likely to effectively solve the problem. This solution should be based on the criteria you’ve established, such as the advantages and disadvantages of each solution, their feasibility and practicability, and your overall goals.

It is critical to remember that there is no one-size-fits-all solution to problems. What is effective for one person or situation may not be effective for another. This is why it is critical to consider a wide range of solutions and evaluate each one based on its ability to effectively solve the problem.

Step 4 – Implement and Monitor the Solution

When you’ve decided on the best solution, it’s time to put it into action. The fourth and final step in effective problem solving is to put the solution into action, monitor its progress, and make any necessary adjustments.

To begin, implement the solution. This may entail delegating tasks, developing a strategy, and allocating resources. Ascertain that everyone involved understands their role and responsibilities in the solution’s implementation.

Next, keep an eye on the solution’s progress. This may entail scheduling regular check-ins, tracking metrics, and soliciting feedback from others. You will be able to identify any potential roadblocks and make any necessary adjustments in a timely manner if you monitor the progress of the solution.

Finally, make any necessary modifications to the solution. This could entail changing the solution, altering the plan of action, or delegating different tasks. Be willing to make changes if they will improve the solution or help it solve the problem more effectively.

It’s important to remember that problem solving is an iterative process, and there may be times when you need to start from scratch. This is especially true if the initial solution does not effectively solve the problem. In these situations, it’s critical to be adaptable and flexible and to keep trying new solutions until you find the one that works best.

To summarise, effective problem solving is a critical skill that can assist individuals and organisations in overcoming challenges and achieving their objectives. Effective problem solving consists of four key steps: defining the problem, generating potential solutions, evaluating alternatives and selecting the best solution, and implementing the solution.

You can increase your chances of success in problem solving by following these steps and considering factors such as the pros and cons of each solution, their feasibility and practicability, and making any necessary adjustments. Furthermore, keep in mind that problem solving is an iterative process, and there may be times when you need to go back to the beginning and restart. Maintain your adaptability and try new solutions until you find the one that works best for you.

• Novick, L.R. and Bassok, M., 2005.  Problem Solving . Cambridge University Press.

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Daniel Croft is a seasoned continuous improvement manager with a Black Belt in Lean Six Sigma. With over 10 years of real-world application experience across diverse sectors, Daniel has a passion for optimizing processes and fostering a culture of efficiency. He's not just a practitioner but also an avid learner, constantly seeking to expand his knowledge. Outside of his professional life, Daniel has a keen Investing, statistics and knowledge-sharing, which led him to create the website learnleansigma.com, a platform dedicated to Lean Six Sigma and process improvement insights.

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Logic and Problem Solving

Problem solving is both an art and science, a truly intellectually entertaining activity. In this course, we will simultaneously bring joy and rigor to solve challenging math problems. We begin by exploring problems from a wide range of difficulty and from a diverse range of areas: logic, combinatorics, inequalities, probability, number theory, geometry, algorithms, and puzzles. We will acquire a toolkit of useful techniques and problem-solving strategies and also develop a disciplined style of thinking that is both creative and logical. There will be ample opportunity to apply these tools and sharpen your thinking skills in solving intriguing problems and puzzles, discuss problem solving with your peers, and present your solutions. Besides elementary algebra, the only prerequisite for this course is your strong craving for fun.

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*The course will meet for two hours daily (Monday–Friday) for a live online class during this window of time. The third hour is used for online office hours. Students will be admitted to and attend just one course section and time. The exact course time and office hour schedule will be set closer to the start of the program. In addition to the live meeting times, students complete out-of-class learning assignments such as assigned readings, group work, pre-recorded online lectures, and more.

Prerequisite(s)

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Critical Thinking and Decision-Making  - What is Critical Thinking?

Critical thinking and decision-making  -, what is critical thinking, critical thinking and decision-making what is critical thinking.

Critical Thinking and Decision-Making: What is Critical Thinking?

Lesson 1: what is critical thinking, what is critical thinking.

Critical thinking is a term that gets thrown around a lot. You've probably heard it used often throughout the years whether it was in school, at work, or in everyday conversation. But when you stop to think about it, what exactly is critical thinking and how do you do it ?

Watch the video below to learn more about critical thinking.

Simply put, critical thinking is the act of deliberately analyzing information so that you can make better judgements and decisions . It involves using things like logic, reasoning, and creativity, to draw conclusions and generally understand things better.

This may sound like a pretty broad definition, and that's because critical thinking is a broad skill that can be applied to so many different situations. You can use it to prepare for a job interview, manage your time better, make decisions about purchasing things, and so much more.

The process

As humans, we are constantly thinking . It's something we can't turn off. But not all of it is critical thinking. No one thinks critically 100% of the time... that would be pretty exhausting! Instead, it's an intentional process , something that we consciously use when we're presented with difficult problems or important decisions.

Improving your critical thinking

In order to become a better critical thinker, it's important to ask questions when you're presented with a problem or decision, before jumping to any conclusions. You can start with simple ones like What do I currently know? and How do I know this? These can help to give you a better idea of what you're working with and, in some cases, simplify more complex issues.  

Real-world applications

Let's take a look at how we can use critical thinking to evaluate online information . Say a friend of yours posts a news article on social media and you're drawn to its headline. If you were to use your everyday automatic thinking, you might accept it as fact and move on. But if you were thinking critically, you would first analyze the available information and ask some questions :

• What's the source of this article?
• Is the headline potentially misleading?
• What are my friend's general beliefs?
• Do their beliefs inform why they might have shared this?

After analyzing all of this information, you can draw a conclusion about whether or not you think the article is trustworthy.

Critical thinking has a wide range of real-world applications . It can help you to make better decisions, become more hireable, and generally better understand the world around you.

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Critical Thinking, Logic & Problem Solving: The Complete Guide to Superior Thinking, Systematic Problem Solving, Making Outstanding Decisions, and Uncover Logical Fallacies Like a Pro Paperback – November 4, 2023

What are the most vital skills to develop in the times we live in? How can you make better decisions?

In the fast-paced world we inhabit, there is a pressing need to cultivate essential skills that are indispensable for success . In an era riddled with fake news, social media, and information overload, the abilities of critical thinking, logic, and problem-solving stand out as the most crucial skills to master . Critical thinking, logic, and problem-solving play pivotal roles in our daily lives , enhancing our ability to t hink effectively and make impeccable decisions . These skills enable us to comprehend the reasons why things are as they are, the influential forces and factors at play, and empower us to develop strategies and alternatives to effectuate change . However, therein lies the problem. Our current education system inadequately, if at all, teaches us these essential skills . This deficiency is the main cause of failure and the underutilization of people's true potential. This Guide was built with the objective of rectifying this problem and equipping you with the most effective tools . This book comprises four key parts: 1) Harnessing the Power of Critical Thinking 2) The Architecture of Thought: Logic, Structuring & Framing 3) The Road to Resolution: Unfolding Problem-Solving 4) Expressing with Impact: The Journey Towards Clear and Effective Communication

Within these pages, you will discover a wealth of knowledge, including: - How to utilize the critical thinking framework - Overcoming mental barriers to critical thinking - Learning to ask the right questions - Detecting fallacies - Recognizing fake news and misinformation - Differentiation and leveraging deductive and inductive reasoning - Leveraging patterns and trends - Exercises to develop critical thinking abilities - Managing emotions to enter the optimal critical thinking zone and mindset - Top key frameworks for framing and describing problems - Utilizing issues trees, logic flow trees, and decision trees like a PRO - The problem-solving process and adopting a scientific mindset - Mastering the assessment of pros and cons - Prioritization matrices - Decision matrices - Organizing action effectively - Leveraging constraints for enhanced creativity - Organizing your communication - Effective storytelling techniques for clear and impactful communication and presentations And much more!

This collection is not merely a theoretical guide. It presents you with the opportunity to practice and effortlessly absorb this powerful knowledge . In no time, you will be capable of constructing compelling arguments, making confident decisions, and identifying any flaws in logic . Critical Thinking, Logic, and Problem-Solving will vastly improve multiple facets of your life, both personal and professional . If you seek the pinnacle of excellence and a fast track to master critical thinking, logic, and problem-solving, look no further.

Don't wait any longer. Grab your copy now and unlock the path to a brighter future! ------------------- ➤ With this book you will also get access to 2 additional exclusive resources: 1) "Unlocking System Thinking" 2) "First Principles Thinking" Discover how inside the book!

• Print length 154 pages
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Join us to rewrite the code of education and unlock the highest potential of human intellect. Ride the wave of discovery and be part of the greatest adventure of all - the pursuit of knowledge.

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26 Expert-Backed Problem Solving Examples – Interview Answers

Published: February 13, 2023

Interview Questions and Answers

Actionable advice from real experts:

Biron Clark

Former Recruiter

Contributor

Dr. Kyle Elliott

Career Coach

Hayley Jukes

Editor-in-Chief

Biron Clark , Former Recruiter

Kyle Elliott , Career Coach

Hayley Jukes , Editor

As a recruiter , I know employers like to hire people who can solve problems and work well under pressure.

 A job rarely goes 100% according to plan, so hiring managers are more likely to hire you if you seem like you can handle unexpected challenges while staying calm and logical.

But how do they measure this?

Hiring managers will ask you interview questions about your problem-solving skills, and they might also look for examples of problem-solving on your resume and cover letter. 

In this article, I’m going to share a list of problem-solving examples and sample interview answers to questions like, “Give an example of a time you used logic to solve a problem?” and “Describe a time when you had to solve a problem without managerial input. How did you handle it, and what was the result?”

• Problem-solving involves identifying, prioritizing, analyzing, and solving problems using a variety of skills like critical thinking, creativity, decision making, and communication.
• Describe the Situation, Task, Action, and Result ( STAR method ) when discussing your problem-solving experiences.
• Tailor your interview answer with the specific skills and qualifications outlined in the job description.
• Provide numerical data or metrics to demonstrate the tangible impact of your problem-solving efforts.

What are Problem Solving Skills? 

Problem-solving is the ability to identify a problem, prioritize based on gravity and urgency, analyze the root cause, gather relevant information, develop and evaluate viable solutions, decide on the most effective and logical solution, and plan and execute implementation. 

Problem-solving encompasses other skills that can be showcased in an interview response and your resume. Problem-solving skills examples include:

• Critical thinking
• Analytical skills
• Decision making
• Research skills
• Technical skills
• Communication skills
• Adaptability and flexibility

Why is Problem Solving Important in the Workplace?

Problem-solving is essential in the workplace because it directly impacts productivity and efficiency. Whenever you encounter a problem, tackling it head-on prevents minor issues from escalating into bigger ones that could disrupt the entire workflow. 

Beyond maintaining smooth operations, your ability to solve problems fosters innovation. It encourages you to think creatively, finding better ways to achieve goals, which keeps the business competitive and pushes the boundaries of what you can achieve. 

Effective problem-solving also contributes to a healthier work environment; it reduces stress by providing clear strategies for overcoming obstacles and builds confidence within teams. 

Examples of Problem-Solving in the Workplace

• Correcting a mistake at work, whether it was made by you or someone else
• Overcoming a delay at work through problem solving and communication
• Resolving an issue with a difficult or upset customer
• Overcoming issues related to a limited budget, and still delivering good work through the use of creative problem solving
• Overcoming a scheduling/staffing shortage in the department to still deliver excellent work
• Troubleshooting and resolving technical issues
• Handling and resolving a conflict with a coworker
• Solving any problems related to money, customer billing, accounting and bookkeeping, etc.
• Taking initiative when another team member overlooked or missed something important
• Taking initiative to meet with your superior to discuss a problem before it became potentially worse
• Solving a safety issue at work or reporting the issue to those who could solve it
• Using problem solving abilities to reduce/eliminate a company expense
• Finding a way to make the company more profitable through new service or product offerings, new pricing ideas, promotion and sale ideas, etc.
• Changing how a process, team, or task is organized to make it more efficient
• Using creative thinking to come up with a solution that the company hasn’t used before
• Performing research to collect data and information to find a new solution to a problem
• Boosting a company or team’s performance by improving some aspect of communication among employees
• Finding a new piece of data that can guide a company’s decisions or strategy better in a certain area

Problem-Solving Examples for Recent Grads/Entry-Level Job Seekers

• Coordinating work between team members in a class project
• Reassigning a missing team member’s work to other group members in a class project
• Adjusting your workflow on a project to accommodate a tight deadline
• Speaking to your professor to get help when you were struggling or unsure about a project
• Asking classmates, peers, or professors for help in an area of struggle
• Talking to your academic advisor to brainstorm solutions to a problem you were facing
• Researching solutions to an academic problem online, via Google or other methods
• Using problem solving and creative thinking to obtain an internship or other work opportunity during school after struggling at first

How To Answer “Tell Us About a Problem You Solved”

When you answer interview questions about problem-solving scenarios, or if you decide to demonstrate your problem-solving skills in a cover letter (which is a good idea any time the job description mentions problem-solving as a necessary skill), I recommend using the STAR method.

STAR stands for:

It’s a simple way of walking the listener or reader through the story in a way that will make sense to them. 

Start by briefly describing the general situation and the task at hand. After this, describe the course of action you chose and why. Ideally, show that you evaluated all the information you could given the time you had, and made a decision based on logic and fact. Finally, describe the positive result you achieved.

Note: Our sample answers below are structured following the STAR formula. Be sure to check them out!

EXPERT ADVICE

Dr. Kyle Elliott , MPA, CHES Tech & Interview Career Coach caffeinatedkyle.com

How can I communicate complex problem-solving experiences clearly and succinctly?

Before answering any interview question, it’s important to understand why the interviewer is asking the question in the first place.

When it comes to questions about your complex problem-solving experiences, for example, the interviewer likely wants to know about your leadership acumen, collaboration abilities, and communication skills, not the problem itself.

Therefore, your answer should be focused on highlighting how you excelled in each of these areas, not diving into the weeds of the problem itself, which is a common mistake less-experienced interviewees often make.

Tailoring Your Answer Based on the Skills Mentioned in the Job Description

As a recruiter, one of the top tips I can give you when responding to the prompt “Tell us about a problem you solved,” is to tailor your answer to the specific skills and qualifications outlined in the job description. 

Once you’ve pinpointed the skills and key competencies the employer is seeking, craft your response to highlight experiences where you successfully utilized or developed those particular abilities. 

For instance, if the job requires strong leadership skills, focus on a problem-solving scenario where you took charge and effectively guided a team toward resolution. 

By aligning your answer with the desired skills outlined in the job description, you demonstrate your suitability for the role and show the employer that you understand their needs.

Amanda Augustine expands on this by saying:

“Showcase the specific skills you used to solve the problem. Did it require critical thinking, analytical abilities, or strong collaboration? Highlight the relevant skills the employer is seeking.”  

Interview Answers to “Tell Me About a Time You Solved a Problem”

Now, let’s look at some sample interview answers to, “Give me an example of a time you used logic to solve a problem,” or “Tell me about a time you solved a problem,” since you’re likely to hear different versions of this interview question in all sorts of industries.

The example interview responses are structured using the STAR method and are categorized into the top 5 key problem-solving skills recruiters look for in a candidate.

1. Analytical Thinking

Situation: In my previous role as a data analyst , our team encountered a significant drop in website traffic.

Task: I was tasked with identifying the root cause of the decrease.

Action: I conducted a thorough analysis of website metrics, including traffic sources, user demographics, and page performance. Through my analysis, I discovered a technical issue with our website’s loading speed, causing users to bounce. 

Result: By optimizing server response time, compressing images, and minimizing redirects, we saw a 20% increase in traffic within two weeks.

2. Critical Thinking

Situation: During a project deadline crunch, our team encountered a major technical issue that threatened to derail our progress.

Task: My task was to assess the situation and devise a solution quickly.

Action: I immediately convened a meeting with the team to brainstorm potential solutions. Instead of panicking, I encouraged everyone to think outside the box and consider unconventional approaches. We analyzed the problem from different angles and weighed the pros and cons of each solution.

Result: By devising a workaround solution, we were able to meet the project deadline, avoiding potential delays that could have cost the company $100,000 in penalties for missing contractual obligations.

3. Decision Making

Situation: As a project manager , I was faced with a dilemma when two key team members had conflicting opinions on the project direction.

Task: My task was to make a decisive choice that would align with the project goals and maintain team cohesion.

Action: I scheduled a meeting with both team members to understand their perspectives in detail. I listened actively, asked probing questions, and encouraged open dialogue. After carefully weighing the pros and cons of each approach, I made a decision that incorporated elements from both viewpoints.

Result: The decision I made not only resolved the immediate conflict but also led to a stronger sense of collaboration within the team. By valuing input from all team members and making a well-informed decision, we were able to achieve our project objectives efficiently.

4. Communication (Teamwork)

Situation: During a cross-functional project, miscommunication between departments was causing delays and misunderstandings.

Task: My task was to improve communication channels and foster better teamwork among team members.

Action: I initiated regular cross-departmental meetings to ensure that everyone was on the same page regarding project goals and timelines. I also implemented a centralized communication platform where team members could share updates, ask questions, and collaborate more effectively.

Result: Streamlining workflows and improving communication channels led to a 30% reduction in project completion time, saving the company $25,000 in operational costs.

5. Persistence 

Situation: During a challenging sales quarter, I encountered numerous rejections and setbacks while trying to close a major client deal.

Task: My task was to persistently pursue the client and overcome obstacles to secure the deal.

Action: I maintained regular communication with the client, addressing their concerns and demonstrating the value proposition of our product. Despite facing multiple rejections, I remained persistent and resilient, adjusting my approach based on feedback and market dynamics.

Result: After months of perseverance, I successfully closed the deal with the client. By closing the major client deal, I exceeded quarterly sales targets by 25%, resulting in a revenue increase of $250,000 for the company.

Tips to Improve Your Problem-Solving Skills

Throughout your career, being able to showcase and effectively communicate your problem-solving skills gives you more leverage in achieving better jobs and earning more money .

So to improve your problem-solving skills, I recommend always analyzing a problem and situation before acting.

 When discussing problem-solving with employers, you never want to sound like you rush or make impulsive decisions. They want to see fact-based or data-based decisions when you solve problems.

Don’t just say you’re good at solving problems. Show it with specifics. How much did you boost efficiency? Did you save the company money? Adding numbers can really make your achievements stand out.

To get better at solving problems, analyze the outcomes of past solutions you came up with. You can recognize what works and what doesn’t.

Think about how you can improve researching and analyzing a situation, how you can get better at communicating, and deciding on the right people in the organization to talk to and “pull in” to help you if needed, etc.

Finally, practice staying calm even in stressful situations. Take a few minutes to walk outside if needed. Step away from your phone and computer to clear your head. A work problem is rarely so urgent that you cannot take five minutes to think (with the possible exception of safety problems), and you’ll get better outcomes if you solve problems by acting logically instead of rushing to react in a panic.

You can use all of the ideas above to describe your problem-solving skills when asked interview questions about the topic. If you say that you do the things above, employers will be impressed when they assess your problem-solving ability.

More Interview Resources

• 3 Answers to “How Do You Handle Stress?”
• How to Answer “How Do You Handle Conflict?” (Interview Question)
• Sample Answers to “Tell Me About a Time You Failed”

About the Author

Biron Clark is a former executive recruiter who has worked individually with hundreds of job seekers, reviewed thousands of resumes and LinkedIn profiles, and recruited for top venture-backed startups and Fortune 500 companies. He has been advising job seekers since 2012 to think differently in their job search and land high-paying, competitive positions. Follow on Twitter and LinkedIn .

Read more articles by Biron Clark

About the Contributor

Kyle Elliott , career coach and mental health advocate, transforms his side hustle into a notable practice, aiding Silicon Valley professionals in maximizing potential. Follow Kyle on LinkedIn .

About the Editor

Hayley Jukes is the Editor-in-Chief at CareerSidekick with five years of experience creating engaging articles, books, and transcripts for diverse platforms and audiences.

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Answering Problem-Solving Interview Questions: Tips and Examples

Problem-solving skills are difficult to describe and quantify: they’re a combination of different hard and soft skills such as logical inference, technical knowledge, adaptability and innovation, leadership potential, decision-making, productivity, and collaboration.

All are crucial for developing expertise and delivering results at work — especially when the going gets tough.

And because problem-solving is so important, you’re almost guaranteed to get asked about it in a job interview. Read on, and make sure no problem-solving question catches you off guard.

In this article, you’ll learn:

• How to answer problem-solving job interview questions
• Types of problem-solving questions
• Why recruiters ask these questions and what your answers might reveal
• Sample answers for the main types of problem-solving questions

Want to get an offer after every interview? Our interview prep tool will guide you through all the questions you can expect, let you record and analyze your answers, and provide instant AI feedback. You’ll know exactly what to improve to turn your next interview into a job.

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How to Answer Problem-Solving Job Interview Questions

Here’s how to understand the intention behind problem-solving interview questions and create an informative answer that will highlight your expertise and potential.

Understand the problem-solving question and why recruiters ask it

Hiring managers and recruiters want to know how you identify roadblocks, analyze information, and overcome challenges. These challenges can vary from specific, technical issues to more general issues like improving company processes or handling interpersonal relationships.

To put these skills to the test, recruiters use “problem-solving” job interview questions, also known as analytical questions. Here are some common ones:

• Tell me about a situation where you had to solve a difficult problem.
• Give me a specific example of a time when you used good judgment and logic in solving a problem.
• Describe a time when you didn’t know how to solve a problem. What did you do?
• Describe how you approach a complex or difficult problem.

Here’s what these questions help recruiters discover:

Your adaptability and innovation

Are you an out-of-the-box thinker who’s open to new ideas and who can handle uncharted waters easily?

Efficiency and productivity

Are your problem-solving skills contributing to the team’s performance, removing bottlenecks, smoothing out processes, and keeping projects on track?

Collaboration and communication

Are you successfully collaborating with others to find solutions? Are you handling people-related problems effectively?

Decision-making

Can you efficiently evaluate different options and reach a decision independently? Can you make sound decisions to minimize risks and maximize benefits and opportunities?

Leadership potential

Are your problem-solving skills so good that they open up new opportunities for you to move in the leadership direction ?

Problem-solving interview questions are not tied to a specific role and industry. Mastering your problem-solving skills will help you stand out from the competition and be more successful in your role, whatever it may be.

And if you need help with answering other common interview questions, sign up for our free course !

Reflect on your thought process

Be mindful of your thought processes when you face a difficult problem.

Is your initial reaction to panic or are you calm and enthusiastic to tackle it? Is the problem stopping you from focusing on everything else you’re working on? Do you look at the problem as a whole or do you break it down?

Understanding how you think and approach the problem will help you know yourself and improve your problem-solving skills, but it’ll also make it easier to answer these tricky questions during an interview.

Be specific

Tailor your answers to problem-solving interview questions so that you cover specific details, actions, and skills relevant to the position. If possible, list the results and share lessons learned from an experience you’re describing.

We’re not saying you should lie and make up a story about your problem-solving skills for each position you apply for; remember that this is a broad set of skills and you surely have something relevant from your past experience that you can bring up.

💡 For example, if you’re a Customer Service Representative applying for the same role in another company, you can speak about how you solved a customer’s problem or how you helped the team switch to a new CRM tool and transfer all the data.

💡 If you’re applying for a leadership role in the customer service field, you can speak about how you handled an interpersonal problem within a team or how you spotted bottlenecks and modified processes to make the team more efficient.

💡 If you’re moving to a Sales position, you can highlight your selling experience and talk about a time when you had to solve a customer’s problem and you managed to upsell them in the process.

Follow up with clear outcomes

Prove you have outstanding problem-solving skills by listing clear outcomes for every problem you solved. They can be quantitative or qualitative.

💡 Fixed a process? Say that it improved team productivity by X%.

💡 Handled a difficult client? If they became a VIP customer later on, mention it.

💡 Resolved a conflict? Describe how the experience helped you strengthen the bonds in a team.

💡 Solved a complex technical problem? Say that you got a bonus for it, or that you expanded and improved the existing documentation to help coworkers in the future.

Use the STAR method

Whenever possible, use the STAR (situation-task-action-result) method in your answer:

• (S) ituation: Describe the situation and provide context.
• (T) ask: What tasks you planned on doing to tackle the issue, your contribution.
• (A) ction you took (step-by-step).
• (R) esult of your efforts.

It’ll help you create a well-rounded answer that’s informative and engaging. Plus, using this method to prepare answers in advance will help you memorize the story quickly and easily.

✅ Bear in mind that not every problem-solving interview question can be answered with a STAR method. Some questions will be very specific and will ask for quick and short information about a certain tool or similar. Other questions, the ones beginning with “Give me an example when…” or “Tell me about a time when…” will be the perfect opportunity to use the STAR method.

Also, remember that there’s never a single correct answer to a problem-solving question, just like there usually are multiple solutions to a given problem — a study on the hospitality industry revealed that the most successful problem-solving strategies applied in the workplace were always very specific to given circumstances.

Questions about your problem-solving skills are just one group of the standard interview questions, you can be almost sure you will get asked. Prepare for other interview “classics” with our dedicated guides:

• Tell Me About Yourself: Sample Answers
• Where Do You See Yourself in 5 Years?
• Why Did You Leave Your Last Job?
• What Are Your Strenghts?
• What Is Your Greatest Weakness?
• How Do You Handle Conflict?
• Why Should We Hire You?
• Why Do You Want to Work Here?

If prepping for a video interview, learn what to expect from this guide: Video Interviewing Tips & Tricks

And if you’re interested in interviewing for specific positions, see:

• Sales Interview Questions and Answers
• Customer Service Interview Questions and Answers
• Customer Service Manager Interview Questions and Answers
• Behavioral Interview Questions for Customer Service

Types of Problem-Solving Job Interview Questions

1. general problem-solving questions.

These questions aim to discover your general approach to problems and challenges.

How do you approach complex problems?

Interviewers want to know how you approach the process of solving complex problems. Do you jump straight into it or do you take a step back, break the problem down into manageable components, analyze the info you have, and then dive in?

Can you provide an example of a challenging issue you’ve encountered and how you resolved it?

Can you assess a situation and find the most appropriate solution? Can you handle the pressure? Do you take the lead during difficult times? Are you able to take responsibility for the outcomes?

This question is more specific than the previous one, so make sure you think about a situation in advance and prepare your answer using the STAR method.

Big Interview’s Answer Builder can help you shape your answer. You’ll be able to list and filter the points you’d like to mention, add details and rearrange the order to create a compelling story.

Plus, you’ll get bite-sized tips on how to answer the most common interview questions while you’re in the Builder.

How do you prioritize multiple tasks when faced with tight deadlines?

Recruiters want to know how you set criteria based on which you’ll set priorities, how and if you juggle between multiple tasks, and how you communicate and collaborate with other people involved.

General problem-solving sample answer

“Tell me about a time when you faced a difficult problem at work. How did you solve it?”

Behavioral questions about problem-solving

Behavioral questions ask for specific situations from your past in which you displayed a certain behavior. Based on it, recruiters hope to predict how you’ll perform in the future.

Tell me about a time when your team faced a problem and you helped to find a solution

This one’s asked to assess your teamwork and cooperation skills in tough situations.

Interestingly, a 2015 study on problem-solving in the workplace showed that when it comes to expertise-related problems, employees rarely relied on trial-and-error or information retrieval as modes of problem-solving.

Instead, they mostly relied on help from others, that is, their coworkers who they believed were experts on the subject matter.

This puts emphasis on the importance of teamwork and collaboration in problem-solving. And you certainly noticed how easier it gets to solve a problem (or brainstorm a new idea) as a group, when different individuals bring fresh, unique ideas to the table.

So, recruiters want to know if you’d be cooperative and open to a teamwork experience, and these factors might hint at how you’ll fit in with the team.

Describe a situation in which you received criticism for your solution to a problem. How did you handle that?

This one checks how you handle feedback and criticism — it’s challenging, but it’s essential for growth.

In your answer, make sure you depict a situation in which you demonstrated that growth mindset and the ability to see that taking criticism is not a sign of weakness (or a personal attack on you) but a unique opportunity to learn something new.

Can you provide an example of when you had to collaborate with a team to solve a work-related problem?

Similarly to the first question in this group, this one aims to see how you perform in a team and solve problems collectively.

According to a study , in a team, task completion can be independent , when each team member completes their own activities, sequential , when activities go from one team member to another, reciprocal , when activities are done back-and-forth between team members, or intensive , when all team members work on activities and problem-solving simultaneously.

Recruiters want to get to know more about your ideal teamwork process model and how you connect with others to solve problems.

Your answer will tell them if you’re a good team problem-solver, team player, and if you’re able to give and share credit, as well as take responsibility if something goes wrong.

Behavioral problem-solving sample answer

“Can you describe a situation where you had to use your problem-solving skills to make a decision?”

Situational problem-solving questions

Situational problem-solving questions put you in a hypothetical situation, present a problem, and ask for your opinion/solution.

Even if you haven’t encountered a similar situation in the past, it will help you to draw parallels from your experience to create answers to these questions.

How would you respond if a high-priority project was suddenly delayed, jeopardizing the deadline?

Your answer to this question will tell recruiters about your flexibility, time and task organization, prioritization, as well as how you handle pressure.

An ideal employee will be able to think quickly and adapt to unforeseen circumstances, all the while remaining calm and composed. You’ll want to aim at displaying these qualities in your answer.

Imagine a scenario where your manager was unavailable, but a client had an urgent issue – what would you do?

Taking the lead and taking calculated risks shows that a person has outstanding problem-solving skills and is not afraid to take initiative, which shows leadership potential.

Your answer to this question needs to demonstrate your ability to quickly analyze information, weigh pros and cons of a situation, and make decisions on the spot. This is especially important if you’re applying for leadership positions, like a team leader or a project manager.

If you encountered a high-stress situation that required you to stay calm and focused, how would you handle it?

Recruiters and hiring managers want to assess your ability to handle stress, make rational decisions, and maintain a focused approach in tricky, high-pressure situations.

Make sure to provide them with relevant examples from your past that will paint a picture of your skills and abilities. This is especially relevant for high-pressure positions such as police officers, lawyers, financial analysts, and similar.

Situational problem-solving sample answer

“Imagine you’re faced with a tight deadline, but you’ve encountered a significant roadblock. How would you handle this situation?”

Technical questions about problem-solving

Technical problem-solving questions are based on the technical knowledge that underlies each role. They aim to check your expertise or the means by which you connect the dots or obtain information if you don’t possess it.

Will you sort through the documentation to find a solution? Or is your first reaction to recall a past experience? Perhaps you prefer connecting with an expert or a coworker with more experience than you. Or you’re the type of person to synthesize your existing knowledge and try to find a solution through trial and error. Maybe you’ll turn to a book or a course? Whatever it is, recruiters would like to know.

There are many ways to solve these problems and your preferred strategies will give recruiters insight into how you think and act.

Examples of technical questions about problem-solving are:

• How would you assess and resolve a performance issue in a web application?
• Describe your approach to troubleshooting a networking issue that spans multiple devices.
• How would you approach debugging a piece of software with limited documentation?
• How would you deal with an angry VIP customer if your boss was away?
• What would you do if you noticed a decline in the ROI of your team?

💡 Bear in mind that, with the rapid development of AI, the majority of technical tasks might be overtaken by robots in the future. That’s why it’s important that you work on your non-technical skills, too. Employers are already admitting that problem-solving skills are the second most important skill they’re looking for. For this reason, researchers are working hard to find and develop frameworks for helping people improve their problem-solving capabilities — you can read more about it in this paper on problem-solving skills among graduate engineers .

Technical problem-solving sample answer

“How would you troubleshoot an error in a software product that has been released to customers?”

✅ Pro tip: Practicing in advance is the only way to make sure your answer is flawless! The Mock Interview Tool will help you record your answer and get instant feedback on its quality and delivery. From power words and your pace of speech to “ummm” counter and eye contact, you’ll get help on how to improve in no time!

Our tool helped AJ land his first job in tech and get 7 job offers in the process . “I think Big Interview was super helpful in that aspect of having canned answers for every possible scenario and being in the moment of answering those questions.”, said AJ.

Problem-Solving Interview Questions: Popular Opinions vs. Expert Advice

Now that we covered different types of problem-solving questions and how to answer them, we decided to dive into popular forums and see what job-seekers have to say on this topic. We picked pieces of advice that resonated with the community and confronted them with expert-backed best practices. Let’s see where we stand.

IndianaJones Jr on Reddit said : “If I was an interviewer asking this question, I would expect a personalized answer relevant to yourself, not to specific projects. At least that’s my interpretation.

“What are your experiences in problem-solving?” 

Sample answer: Generally, when I’m working on a project I find it’s easier to start at the end and work backwards. I use that to get a broad strokes idea of where my work needs to take me on any particular project and then I head in that direction. I find that when I get to specific problems I can get too stuck on using tried and true methods so I try to encourage myself to use out-of-the-box solutions. For example [your example here]…”

Career expert comments:

The “bones” of this sample answer are solid. It puts emphasis on breaking down the candidate’s thought process and displays patterns through which the candidate solves problems and learns along the way. However, the most important part of the answer — the actual example of a candidate’s problem-solving skills put to practice — remains a placeholder. Remember, the more specific you get in your answer, the better the impression you make on the interviewer. So here, I recommend paying equal attention to a specific situation in which you solved a problem and using the STAR method to tell that story.

Ambitious_Tell_4852 , when discussing the question “Give an example of a challenge you faced and how you overcame it,” said: 

“Clearly, that is the standard trick question designed for a prospective new hire to tell a prospective employer about his/her professional weaknesses. Oldest “negative Nelli’’ question imaginable during the interview process. Always keep your answer thorough and positive albeit sickeningly sugar-coated! 😁”

This is, straight-out, a bad piece of advice. If an interviewer wants to hear about your weaknesses, they will ask “What is your greatest weakness?” 

A question about overcoming a challenge isn’t a trick question at all. I’d argue it’s actually an opportunity to share some of your proudest wins. But when it comes to answering this question, it’s true that your answers do need to be thorough and positive. This doesn’t mean you need to sugar-coat anything, though. Interviewers don’t want to hear you downplaying your challenges. On the contrary, they want to hear you speak about them honestly and explain what you learned from them. And being able to do so puts a healthy, positive spin on the situation. To put it shortly: provide a real example from your past, answer this question honestly, and emphasize the results and lessons learned. 

Here’s an opinion from a hiring manager, Hugh on Quora, about how to answer a question about a time you needed to solve a problem:

“It really doesn’t matter what the problem you describe is or how you solved it. What I am looking/listening for is 1) the size of the problem (the bigger, the better, a broken shoelace before going out on a date is not an impressive problem) and 2) a step-by-step process to a satisfactory solution (if suddenly all variables fell into place does not show me that you solved the problem — you were just there when it solved itself).

I am also looking/listening for an example of how you solve a problem after you are hired. I may have to explain it to my superiors, and I would like to know that I have a complete and accurate story to tell.”

Career expert comments:  

A good piece of advice from someone who has first-hand hiring experience. When talking about problem-solving, a detailed description of your process is key. The only thing I wouldn’t agree with is having to choose a “big” problem. If you do have experience solving a big problem, that’s great. But sometimes you won’t have a major problem to talk about, and it largely depends on your level of experience and your position. So pick a relevant difficulty, even if it’s not that big, in which you displayed skills relevant to the role you’re applying for.

• Problem-solving skills encompass your logical inference, technical knowledge, adaptability and innovation, leadership potential, decision-making, productivity, and collaboration.
• Because these skills are important in the workplace, there’s a variety of problem-solving interview questions recruiters will ask to assess you.
• Some of them include behavioral, situational, or technical problem-solving questions.
• In order to answer these questions, you need to be aware of your thought processes when faced with a problem.
• In your answer, be as specific as you can and use the STAR format whenever possible.
• Make sure to highlight outcomes, results, or lessons learned.
• As always, the best strategy is to anticipate these questions and prepare rough answers in advance. Including practicing your answer so you’re confident for your interview.

____________________

Need help with your job search? There are 3 ways we can help you:

• Tired of interviewing and not landing the job? Discover actionable lessons and interview practice here (Rated with 4.9/5 by 1,000,000 users).
• Learn how to talk about your proudest accomplishments without bragging or underselling yourself.
• Learn how to answer tricky questions about conflict resolution in the workplace.

How can I improve my problem-solving skills?

Stay in the loop with new technologies and trends. Accept challenges and problems as a way to grow, don’t panic over them. Acquire a systematic approach to analyzing problems, break them down into smaller components which will help you discover root causes and devise a solution plan. Practice logical thinking, evaluating evidence, and staying objective. And give yourself time. Perhaps not surprisingly, studies suggest that the more business experience you have, the better you become at problem-solving.

Are there specific resources available to practice problem-solving interview questions?

There’s a variety of resources available to you, such as courses and Youtube tutorials, Facebook/LinkedIn groups, forums such as Reddit and Quora, books, or online platforms like Big Interview. If you’re trying to develop technical problem-solving skills, you might benefit from relevant platforms’ knowledge bases or YT channels; but if you’re looking specifically for how to answer interview questions, platforms like Big Interview are the way to go.

How should I handle a question about a problem-solving scenario I have not encountered before?

Don’t be afraid to ask additional questions for clarification. If you’ve never dealt with this problem before, be honest about it but answer how you would solve the problem if you were faced with it today. Break the problem down into manageable steps, try to recall a similar situation from your own experience that could help you draw parallels, and propose several different solutions.

Can I talk about my problem-solving experiences derived from non-professional settings such as student projects?

Yes, especially if you’re a recent graduate or a candidate with limited experience. You can use experiences and examples from student projects, extracurricular activities, and you can even use examples from your personal life, as long as you present them in a professional manner and connect them to the position you’re applying for. Remember to highlight the results, as well as the skills that helped you solve the problem and that are relevant to the position you’re applying for.

Are there any common mistakes to avoid when answering problem-solving questions during an interview?

The most common mistake is not preparing in advance which causes rambling. You need to make sure that your answer is informative and well-structured, and that you’re not only presenting a solution but also laying down the steps to display your logical reasoning. Make sure not to forget to give credit to teammates if they contributed to solving the problem you chose to talk about. Finally, for a coherent and informative presentation, make sure you use the STAR method.

What can I do if I don’t know the answer to a technical problem-solving question in an interview?

Handle it professionally. You can always try to reach a conclusion by breaking down the problem and thinking out loud to show your thinking mechanism. Draw parallels between the problem at hand and another similar problem you encountered before. Lay down possible solutions, even if you’re not sure they’ll work, and be transparent — feel free to tell the recruiter you’re not sure how to answer it, but make sure you emphasize that you’re open to learning.

Can I ask for help or guidance from the interviewer during a problem-solving question?

Avoid asking for help directly, but ask for clarification in case something is unclear or if you need additional information. Sometimes, the interviewer will take the initiative and provide you with hints to encourage you and see how you think.

How can I demonstrate creativity and resourcefulness when answering problem-solving questions?

It’s all about storytelling! Preparing in advance will provide some space for displaying your creativity. You can do it by making fun analogies or drawing parallels from well-known situations; or making pop-culture references.

Maja Stojanovic

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What is Cognitive Behavioral Therapy?

Cognitive behavioral therapy (CBT) is a form of psychological treatment that has been demonstrated to be effective for a range of problems including depression, anxiety disorders, alcohol and drug use problems, marital problems, eating disorders, and severe mental illness. Numerous research studies suggest that CBT leads to significant improvement in functioning and quality of life. In many studies, CBT has been demonstrated to be as effective as, or more effective than, other forms of psychological therapy or psychiatric medications.

It is important to emphasize that advances in CBT have been made on the basis of both research and clinical practice. Indeed, CBT is an approach for which there is ample scientific evidence that the methods that have been developed actually produce change. In this manner, CBT differs from many other forms of psychological treatment.

CBT is based on several core principles, including:

• Psychological problems are based, in part, on faulty or unhelpful ways of thinking.
• Psychological problems are based, in part, on learned patterns of unhelpful behavior.
• People suffering from psychological problems can learn better ways of coping with them, thereby relieving their symptoms and becoming more effective in their lives.

CBT treatment usually involves efforts to change thinking patterns. These strategies might include:

• Learning to recognize one’s distortions in thinking that are creating problems, and then to reevaluate them in light of reality.
• Gaining a better understanding of the behavior and motivation of others.
• Using problem-solving skills to cope with difficult situations.
• Learning to develop a greater sense of confidence in one’s own abilities.

CBT treatment also usually involves efforts to change behavioral patterns. These strategies might include:

• Facing one’s fears instead of avoiding them.
• Using role playing to prepare for potentially problematic interactions with others.
• Learning to calm one’s mind and relax one’s body.

Not all CBT will use all of these strategies. Rather, the psychologist and patient/client work together, in a collaborative fashion, to develop an understanding of the problem and to develop a treatment strategy.

CBT places an emphasis on helping individuals learn to be their own therapists. Through exercises in the session as well as “homework” exercises outside of sessions, patients/clients are helped to develop coping skills, whereby they can learn to change their own thinking, problematic emotions, and behavior.

CBT therapists emphasize what is going on in the person’s current life, rather than what has led up to their difficulties. A certain amount of information about one’s history is needed, but the focus is primarily on moving forward in time to develop more effective ways of coping with life.

Source: APA Div. 12 (Society of Clinical Psychology)

Correlation of free-body diagrams to problem-solving skill on particle dynamics topic

• Rizkianto, Dimas Dafa
• Kurniawati, Mega Putri
• Taqwa, Muhammad Reyza Arief

Free Body Diagrams (FBDs) are devices used to abstract force vectors in Newtonian mechanics, which can provide an overview of students' skills in performing force and vector analysis. Problem-solving skills are one of the skills that are needed when identifying, determining and solving problems by using creative thinking, literacy, and logic. Previous research indicated that students had difficulties when solving problems, especially in vector and force analysis, which then had a negative impact on the use of Newton's law concept. Therefore, this study aims to provide an overview of problem-solving skills, FBD quality, and the correlation between FBD quality scores and problem-solving skills. In this study, a descriptive quantitative method was used. The research data was obtained by providing instruments in the form of 5 FBDs essay questions and 5 problem-solving essay questions to physics students in UM who have taken the topic of particle dynamics. The research carried out produced data in the form of the average FBD skill by students of 38.04%, the average problem-solving skill of 65.14%, and the good quality of FBDs correlated with good problem-solving skill with a percentage of 23%.

• PHYSICS EDUCATION